Angles | Free ACT Math Course | The School of Mathematics

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<h1>ACT Math: Angles</h1>

Angles is a foundational Plane Geometry topic on the ACT, and every question tends to boil down to a handful of core relationships once you recognize the pattern. This free, complete lesson covers angle relationships (complementary, supplementary, vertical, and linear pairs), parallel lines cut by a transversal, the triangle angle sum and exterior angle theorem, and polygon interior and exterior angle sums. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-angles-quiz-1">Practice Angles Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#relationships">Angle Relationships</a></li>

<li><a href="#transversals">Parallel Lines and Transversals</a></li>

<li><a href="#triangles">Triangle Angle Sum and the Exterior Angle Theorem</a></li>

<li><a href="#polygons">Polygon Interior and Exterior Angle Sums</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

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<h2 id="relationships">1. Angle Relationships</h2>

<p><strong>Complementary angles</strong> sum to 90°. <strong>Supplementary angles</strong> sum to 180°. When two lines cross, they form two pairs of <strong>vertical angles</strong>, which are always equal, and adjacent pairs called <strong>linear pairs</strong>, which are always supplementary.</p>

<p><strong>Worked Example:</strong> Two angles form a linear pair. One measures 112°. Find the other.</p>

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<p class="prompt">1. Two angles are complementary. One measures 27°. What is the other?</p>

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<p><span class="label">Explanation:</span> 90° − 27° = 63°.</p>

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<p class="prompt">2. Two angles are supplementary. One measures 98°. What is the other?</p>

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<p><span class="label">Explanation:</span> 180° − 98° = 82°.</p>

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<p class="prompt">3. Two lines intersect, forming vertical angles. One measures 47°. What is its vertical angle?</p>

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<p><span class="label">Explanation:</span> Vertical angles are always equal, so the vertical angle also measures 47°.</p>

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<p class="prompt">4. Two angles form a linear pair. One measures 65°. What is the other?</p>

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<p><span class="label">Explanation:</span> A linear pair is always supplementary: 180° − 65° = 115°.</p>

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<p class="prompt">5. Four angles meet at a single point (forming a full circle around it). Three of them measure 80°, 95°, and 110°. What is the measure of the fourth angle?</p>

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<p><span class="label">Explanation:</span> Angles all the way around a single point sum to 360°: 360° − 80° − 95° − 110° = 75°.</p>

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<h2 id="transversals">2. Parallel Lines and Transversals</h2>

<p>When a transversal crosses two parallel lines, it creates eight angles, but only two distinct angle measures appear in the whole figure.</p>

<tr><th>Angle pair</th><th>Relationship</th></tr>

<tr><td>Corresponding angles</td><td>Equal</td></tr>

<tr><td>Alternate interior angles</td><td>Equal</td></tr>

<tr><td>Alternate exterior angles</td><td>Equal</td></tr>

<tr><td>Co-interior (same-side interior) angles</td><td>Supplementary</td></tr>

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<p class="figure-caption">Parallel lines cut by a transversal</p>

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<p><strong>Worked Example:</strong> Two parallel lines are cut by a transversal. One angle measures 72°. Find its co-interior (same-side interior) angle.</p>

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<p class="prompt">1. Two parallel lines are cut by a transversal. One angle measures 58°. What is its corresponding angle?</p>

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<p><span class="label">Explanation:</span> Corresponding angles are equal: 58°.</p>

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<p class="prompt">2. Two parallel lines are cut by a transversal. One angle measures 58°. What is its alternate interior angle?</p>

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<p><span class="label">Explanation:</span> Alternate interior angles are equal: 58°.</p>

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<p class="prompt">3. Two parallel lines are cut by a transversal. One angle measures 58°. What is its co-interior (same-side interior) angle?</p>

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<p><span class="label">Explanation:</span> Co-interior angles are supplementary: 180° − 58° = 122°.</p>

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<p class="prompt">4. Two parallel lines are cut by a transversal. Two co-interior angles measure (3x + 10)° and (2x + 30)°. What is the value of x?</p>

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<p><span class="label">Explanation:</span> Co-interior angles sum to 180°: (3x + 10) + (2x + 30) = 180. This gives 5x + 40 = 180, so 5x = 140 and x = 28.</p>

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<p class="prompt">5. Two parallel lines are cut by a transversal. One angle measures (4x − 10)° and its corresponding angle measures (2x + 30)°. What is the value of x?</p>

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<p><span class="label">Explanation:</span> Corresponding angles are equal: 4x − 10 = 2x + 30. This gives 2x = 40, so x = 20.</p>

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<h2 id="triangles">3. Triangle Angle Sum and the Exterior Angle Theorem</h2>

<p>The interior angles of any triangle always sum to 180°. An <strong>exterior angle</strong>, formed by extending one side, always equals the sum of the two <strong>remote interior angles</strong> (the two angles not adjacent to it).</p>

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<p class="figure-caption">Exterior angle = sum of the two remote interior angles</p>

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<p><strong>Worked Example:</strong> A triangle has angles measuring 2x, 3x, and 4x. Find the measure of each angle.</p>

<p>The angles measure 40°, 60°, and 80°.</p>

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<p class="prompt">1. A triangle has angles of 55° and 65°. What is the measure of the third angle?</p>

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<p><span class="label">Explanation:</span> 180° − 55° − 65° = 60°.</p>

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<p class="prompt">2. An isosceles triangle has a vertex angle of 50°. What is the measure of each base angle?</p>

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<p><span class="label">Explanation:</span> The two base angles of an isosceles triangle are equal. Together they take up 180° − 50° = 130°, so each is 130° ÷ 2 = 65°.</p>

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<p class="prompt">3. A triangle has angles x, 2x, and 3x. What is the measure of the largest angle?</p>

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<p><span class="label">Explanation:</span> x + 2x + 3x = 180, so 6x = 180 and x = 30. The largest angle is 3(30) = 90°.</p>

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<p class="prompt">4. An exterior angle of a triangle measures 115°. What is the sum of the two remote interior angles?</p>

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<p><span class="label">Explanation:</span> By the exterior angle theorem, an exterior angle always equals the sum of the two remote interior angles: 115°.</p>

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<p class="prompt">5. An exterior angle of a triangle measures 130°, and one remote interior angle measures 70°. What is the other remote interior angle?</p>

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<p><span class="label">Explanation:</span> 130° − 70° = 60°.</p>

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<h2 id="polygons">4. Polygon Interior and Exterior Angle Sums</h2>

<p>The sum of the interior angles of any convex polygon with n sides is 180°(n − 2). The sum of the exterior angles of any convex polygon, one at each vertex, is always 360°, regardless of the number of sides.</p>

<p><strong>Worked Example:</strong> Find the sum of the interior angles of a hexagon (6 sides).</p>

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<p class="prompt">1. Find the sum of the interior angles of a pentagon (5 sides).</p>

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<p><span class="label">Explanation:</span> 180°(5 − 2) = 180°(3) = 540°.</p>

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<p class="prompt">2. Each interior angle of a regular polygon measures 144°. How many sides does the polygon have?</p>

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<p><span class="label">Explanation:</span> Each exterior angle is 180° − 144° = 36°. Since the exterior angles of a regular polygon are all equal and sum to 360°, the number of sides is 360° ÷ 36° = 10.</p>

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<p class="prompt">3. What is the sum of the exterior angles of any convex polygon, one at each vertex?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-3',false)">C) Depends on the number of sides</button>

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<p><span class="label">Explanation:</span> No matter how many sides a convex polygon has, its exterior angles (one per vertex) always sum to exactly 360°, a full turn around the shape.</p>

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<p class="prompt">4. Find the measure of each interior angle of a regular octagon (8 sides).</p>

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<p><span class="label">Explanation:</span> Sum of interior angles: 180°(8 − 2) = 1,080°. Since a regular octagon has 8 equal angles: 1,080° ÷ 8 = 135°.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Mixing up complementary and supplementary.</strong> Complementary sums to 90°, supplementary sums to 180°. Confusing the two flips your entire answer.</li>

<li><strong>Treating co-interior angles as equal instead of supplementary.</strong> Corresponding, alternate interior, and alternate exterior angles are equal; co-interior (same-side interior) angles are the one pair that's supplementary instead.</li>

<li><strong>Forgetting that the exterior angle theorem only applies to triangles.</strong> "Exterior angle equals the sum of the two remote interior angles" is specific to triangles; for other polygons, use the general 180°(n − 2) interior angle sum instead.</li>

<li><strong>Using the interior angle sum formula for exterior angles, or vice versa.</strong> Interior angles sum to 180°(n − 2), which depends on the number of sides. Exterior angles always sum to 360°, no matter how many sides the polygon has.</li>

<li><strong>Assuming a polygon is regular without being told so.</strong> The "divide the total by the number of sides" shortcut for finding one interior angle only works when every angle in the polygon is equal.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>Is there a fast way to remember all the parallel-line angle relationships?</h3>

<p>Sort every pair into one of two buckets: angles in matching positions relative to the two intersections (corresponding, alternate interior, alternate exterior) are equal, while angles on the same side, between the parallel lines (co-interior), are supplementary. That single distinction covers the whole topic.</p>

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<h3>Why does the exterior angle of a triangle equal the sum of the remote interior angles?</h3>

<p>The exterior angle and its adjacent interior angle form a linear pair, so together they sum to 180°. Since all three interior angles of the triangle also sum to 180°, the exterior angle must equal whatever's left after removing its adjacent interior angle, which is exactly the sum of the other two.</p>

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<h3>How can I verify the interior angle sum formula for myself?</h3>

<p>Any convex polygon with n sides can be split into (n − 2) triangles by drawing diagonals from one vertex. Since each triangle contributes 180°, the total interior angle sum is 180°(n − 2).</p>

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<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-angles-quiz-1">Angles quizzes</a> in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-angles-quiz-1">Practice Angles Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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