\documentclass[12pt]{article} \usepackage{tikz} \usepackage{amsmath} % Underlining package \usepackage{ulem} \usetikzlibrary{calc} \usepackage[a4paper, portrait, margin=1cm]{geometry} \usepackage{fancyhdr} \def \HeadingQuestions {\section*{\Large Name: \underline{\hspace{8cm}} \hfill Date: \underline{\hspace{3cm}}} \vspace{-3mm} {Angles in a Triangle: Questions} \vspace{1pt}\hrule} % raise footer with page number; no header \fancypagestyle{myfancypagestyle}{ \fancyhf{}% clear all header and footer fields \renewcommand{\headrulewidth}{0pt} % no rule under header \fancyfoot[C] {\thepage} \setlength{\footskip}{14.5pt} % raise page number allowed min 14.5pt } \pagestyle{myfancypagestyle} % apply myfancypagestyle \newcounter{minipagecount} \begin{document} \HeadingQuestions \vspace{8mm} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=268] \coordinate (A) at (0,0); \coordinate (B) at (3.3358087076495626,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(27:4cm)$)}, second line={(B)--($(B)+(180-27:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=27, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=27, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-27, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-27, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+27, end angle=360-27, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {L}; \node at ($(B)!-0.25cm!(C)$) {M}; \node at ($(C)!-0.25cm!(A)$) {N}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$27^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=317] \coordinate (A) at (0,0); \coordinate (B) at (3.014764869491839,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(53:4cm)$)}, second line={(B)--($(B)+(180-53:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=53, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=53, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-53, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-53, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+53, end angle=360-53, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {A}; \node at ($(B)!-0.25cm!(C)$) {B}; \node at ($(C)!-0.25cm!(A)$) {C}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$\theta^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$74^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=314] \coordinate (A) at (0,0); \coordinate (B) at (3.698829928261037,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(58:4cm)$)}, second line={(B)--($(B)+(180-58:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=58, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=58, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-58, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-58, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+58, end angle=360-58, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {G}; \node at ($(B)!-0.25cm!(C)$) {F}; \node at ($(C)!-0.25cm!(A)$) {H}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$\theta^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$64^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=202] \coordinate (A) at (0,0); \coordinate (B) at (3.282422598576085,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(61:4cm)$)}, second line={(B)--($(B)+(180-61:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=61, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=61, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-61, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-61, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+61, end angle=360-61, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {R}; \node at ($(B)!-0.25cm!(C)$) {T}; \node at ($(C)!-0.25cm!(A)$) {S}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$\theta^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$58^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=217] \coordinate (A) at (0,0); \coordinate (B) at (3.5421165755066077,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(51:4cm)$)}, second line={(B)--($(B)+(180-51:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=51, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=51, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-51, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-51, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+51, end angle=360-51, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {A}; \node at ($(B)!-0.25cm!(C)$) {B}; \node at ($(C)!-0.25cm!(A)$) {C}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$\theta^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$78^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=199] \coordinate (A) at (0,0); \coordinate (B) at (3.0664161685590976,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(36:4cm)$)}, second line={(B)--($(B)+(180-36:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=36, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=36, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-36, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-36, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+36, end angle=360-36, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {M}; \node at ($(B)!-0.25cm!(C)$) {L}; \node at ($(C)!-0.25cm!(A)$) {N}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$36^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=301] \coordinate (A) at (0,0); \coordinate (B) at (3.550467202902078,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(62:4cm)$)}, second line={(B)--($(B)+(180-62:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=62, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=62, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-62, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-62, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+62, end angle=360-62, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {X}; \node at ($(B)!-0.25cm!(C)$) {Z}; \node at ($(C)!-0.25cm!(A)$) {Y}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$\theta^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$56^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=324] \coordinate (A) at (0,0); \coordinate (B) at (3.326476120781351,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(46:4cm)$)}, second line={(B)--($(B)+(180-46:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=46, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=46, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-46, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-46, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+46, end angle=360-46, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {C}; \node at ($(B)!-0.25cm!(C)$) {A}; \node at ($(C)!-0.25cm!(A)$) {B}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$46^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=255] \coordinate (A) at (0,0); \coordinate (B) at (3.2381276565662915,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(43:4cm)$)}, second line={(B)--($(B)+(180-43:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=43, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=43, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-43, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-43, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+43, end angle=360-43, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {C}; \node at ($(B)!-0.25cm!(C)$) {B}; \node at ($(C)!-0.25cm!(A)$) {A}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$43^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=241] \coordinate (A) at (0,0); \coordinate (B) at (3.4649480770730365,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(64:4cm)$)}, second line={(B)--($(B)+(180-64:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=64, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=64, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-64, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-64, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+64, end angle=360-64, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {C}; \node at ($(B)!-0.25cm!(C)$) {B}; \node at ($(C)!-0.25cm!(A)$) {A}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$64^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=3] \coordinate (A) at (0,0); \coordinate (B) at (3.6248019295345557,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(55:4cm)$)}, second line={(B)--($(B)+(180-55:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=55, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=55, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-55, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-55, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+55, end angle=360-55, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {S}; \node at ($(B)!-0.25cm!(C)$) {T}; \node at ($(C)!-0.25cm!(A)$) {R}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$55^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=58] \coordinate (A) at (0,0); \coordinate (B) at (3.4533808108364052,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(48:4cm)$)}, second line={(B)--($(B)+(180-48:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=48, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=48, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-48, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-48, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+48, end angle=360-48, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {R}; \node at ($(B)!-0.25cm!(C)$) {T}; \node at ($(C)!-0.25cm!(A)$) {S}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$48^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=25] \coordinate (A) at (0,0); \coordinate (B) at (3.606909340979945,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(61:4cm)$)}, second line={(B)--($(B)+(180-61:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=61, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=61, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-61, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-61, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+61, end angle=360-61, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {B}; \node at ($(B)!-0.25cm!(C)$) {A}; \node at ($(C)!-0.25cm!(A)$) {C}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$\theta^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$58^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=312] \coordinate (A) at (0,0); \coordinate (B) at (3.2778891459611605,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(44:4cm)$)}, second line={(B)--($(B)+(180-44:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=44, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=44, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-44, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-44, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+44, end angle=360-44, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {C}; \node at ($(B)!-0.25cm!(C)$) {B}; \node at ($(C)!-0.25cm!(A)$) {A}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$\theta^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$92^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=58] \coordinate (A) at (0,0); \coordinate (B) at (3.3414709886830494,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(54:4cm)$)}, second line={(B)--($(B)+(180-54:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=54, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=54, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-54, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-54, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+54, end angle=360-54, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {L}; \node at ($(B)!-0.25cm!(C)$) {M}; \node at ($(C)!-0.25cm!(A)$) {N}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$\theta^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$72^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=268] \coordinate (A) at (0,0); \coordinate (B) at (3.989110756926936,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(59:4cm)$)}, second line={(B)--($(B)+(180-59:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=59, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=59, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-59, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-59, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+59, end angle=360-59, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {N}; \node at ($(B)!-0.25cm!(C)$) {M}; \node at ($(C)!-0.25cm!(A)$) {L}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$59^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=60] \coordinate (A) at (0,0); \coordinate (B) at (3.4664015205586622,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(25:4cm)$)}, second line={(B)--($(B)+(180-25:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=25, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=25, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-25, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-25, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+25, end angle=360-25, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {C}; \node at ($(B)!-0.25cm!(C)$) {B}; \node at ($(C)!-0.25cm!(A)$) {A}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$\theta^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$130^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=120] \coordinate (A) at (0,0); \coordinate (B) at (3.995682249069551,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(37:4cm)$)}, second line={(B)--($(B)+(180-37:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=37, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=37, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-37, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-37, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+37, end angle=360-37, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {Z}; \node at ($(B)!-0.25cm!(C)$) {X}; \node at ($(C)!-0.25cm!(A)$) {Y}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$37^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=294] \coordinate (A) at (0,0); \coordinate (B) at (3.1958293459108207,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(41:4cm)$)}, second line={(B)--($(B)+(180-41:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=41, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=41, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-41, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-41, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+41, end angle=360-41, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {R}; \node at ($(B)!-0.25cm!(C)$) {S}; \node at ($(C)!-0.25cm!(A)$) {T}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {$\theta^\circ$}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$98^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= \frac{(180^\circ - \angle \text{\dotuline{~~~~~~~}})}{2} \\ &= \frac{(180^\circ - \dotuline{~~~~~~~}^\circ)}{2} \\ &= \frac{\dotuline{~~~~~~~}^\circ}{2} \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \begin{minipage}{0.55\textwidth} \refstepcounter{minipagecount} \noindent{(\theminipagecount)}\quad \begin{tikzpicture}[scale=1.5, baseline=(current bounding box.north)] \begin{scope}[rotate=14] \coordinate (A) at (0,0); \coordinate (B) at (3.1219129629741484,0); \coordinate (C) at (intersection cs: first line={(A)--($(A)+(25:4cm)$)}, second line={(B)--($(B)+(180-25:4cm)$)}); \draw (A) -- (B) -- (C) -- cycle; % Mark angles with arcs; double arc for equal angles \draw ($(A)!0.23cm!(B)$) arc [start angle=0, end angle=25, radius=0.23cm]; \draw ($(A)!0.3cm!(B)$) arc [start angle=0, end angle=25, radius=0.3cm]; \draw ($(B)!0.23cm!(C)$) arc [start angle=180-25, end angle=180, radius=0.23cm]; \draw ($(B)!0.3cm!(C)$) arc [start angle=180-25, end angle=180, radius=0.3cm]; \draw ($(C)!0.3cm!(A)$) arc [start angle=180+25, end angle=360-25, radius=0.3cm]; % Label angles \node at ($(A)!-0.25cm!(B)$) {F}; \node at ($(B)!-0.25cm!(C)$) {G}; \node at ($(C)!-0.25cm!(A)$) {H}; % Mark angles in degrees \coordinate (midBC) at ($(B)!0.5!(C)$); \node at ($(A)!0.55cm!(midBC)$) {$25^\circ$}; \coordinate (midAC) at ($(A)!0.5!(C)$); \node at ($(B)!0.55cm!(midAC)$) {}; \coordinate (midAB) at ($(A)!0.5!(B)$); \node at ($(C)!0.55cm!(midAB)$) {$\theta^\circ$}; % Draw hash mark perpendicular to line AB at its midpoint \draw[black] ($(midAC)!1.5mm!90:(A)$)--($(midAC)!1.5mm!-90:(A)$); % Add hash marks on side AC \draw[black] ($(midBC)!1.5mm!90:(B)$)--($(midBC)!1.5mm!-90:(B)$); \end{scope} \end{tikzpicture} \end{minipage}% \hfill \begin{minipage}{0.4\textwidth} \begin{align*} \text{$\theta^\circ$} &= 180^\circ - (\angle \text{\dotuline{~~~~~~~}} + \angle \text{\dotuline{~~~~~~~}}) \\ &= 180^\circ - (\dotuline{~~~~~~~}^\circ + \dotuline{~~~~~~~}^\circ) \\ &= 180^\circ - \dotuline{~~~~~~~}^\circ \\ &= \dotuline{~~~~~~~}^\circ \end{align*} \end{minipage} \end{document}