Lines, Angles & Triangles | Free SAT Course

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<title>Lines, Angles, and Triangles - Free SAT Math Lesson & Practice | The School of Mathematics</title>

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<h1>Lines, Angles, and Triangles: Free SAT Math Lesson</h1>

"Lines, Angles, and Triangles" builds the geometric foundation for nearly everything else in the Geometry and Trigonometry domain of the Digital SAT. This free, complete lesson covers complementary, supplementary, and vertical angle relationships, the angles formed when parallel lines are cut by a transversal, the triangle angle sum and exterior angle theorem, the criteria for proving triangles congruent or similar, and the triangle inequality theorem. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/sat-lines-angles-triangles-quiz-1">Practice Lines, Angles & Triangles Free</a>

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

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

<ol>

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

<li><a href="#triangle-sum">Triangle Angle Sum and Exterior Angles</a></li>

<li><a href="#congruence">Triangle Congruence and Similarity</a></li>

<li><a href="#inequality">The Triangle Inequality</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="angles">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.</p>

<p>When two parallel lines are cut by a transversal, eight angles are formed, and several useful relationships appear:</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">Two parallel lines (ℓ₁, ℓ₂) cut by a transversal, forming 8 angles</p>

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<p><strong>Worked Example:</strong> Two parallel lines are cut by a transversal. One angle measures 65°. Find its corresponding angle and its co-interior (same-side interior) angle.</p>

<p>Corresponding angles are equal, so the corresponding angle also measures 65°.</p>

<p>Co-interior angles are supplementary, so that angle measures 180° − 65° = 115°.</p>

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<p class="prompt">1. Two angles are complementary. One measures 37°. What is the measure of the other?</p>

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<p><span class="label">Explanation:</span> Complementary angles sum to 90°: 90° − 37° = 53°.</p>

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<p class="prompt">2. Two angles are supplementary. One measures 112°. What is the measure of the other?</p>

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<p><span class="label">Explanation:</span> Supplementary angles sum to 180°: 180° − 112° = 68°.</p>

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<p class="prompt">3. Two lines intersect, forming vertical angles. If one angle measures 74°, what is the measure of 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 74°.</p>

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<p class="prompt">4. Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of its alternate interior angle?</p>

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<p><span class="label">Explanation:</span> Alternate interior angles are equal, so this angle also measures 65°.</p>

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<p class="prompt">5. Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of its co-interior (same-side interior) angle?</p>

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<p><span class="label">Explanation:</span> This is the worked example above: co-interior angles are supplementary, so 180° − 65° = 115°.</p>

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<h2 id="triangle-sum">2. Triangle Angle Sum and Exterior Angles</h2>

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

<text x="160" y="122" font-size="10" fill="#ea4335">exterior &angle;</text>

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<p class="figure-caption">Exterior angle at C equals the sum of remote interior angles A and B</p>

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<p><strong>Worked Example:</strong> A triangle has angles measuring 3x, 4x, and 5x. Find the measure of each angle.</p>

<p>The angles measure 3(15) = 45°, 4(15) = 60°, and 5(15) = 75°.</p>

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<p class="prompt">1. A triangle has angles measuring 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">2. A triangle has two angles measuring 48° and 67°. What is the measure of the third angle?</p>

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<p><span class="label">Explanation:</span> 180° − 48° − 67° = 65°.</p>

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<p class="prompt">3. An exterior angle of a triangle measures 110°. The two remote interior angles are equal. What is the measure of each?</p>

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<p><span class="label">Explanation:</span> The exterior angle equals the sum of the two remote interior angles: 110° ÷ 2 = 55° each.</p>

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<p class="prompt">4. A triangle has one angle of 90° and another of 35°. What is the measure of the third angle?</p>

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<p><span class="label">Explanation:</span> 180° − 90° − 35° = 55°.</p>

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<h2 id="congruence">3. Triangle Congruence and Similarity</h2>

<p><strong>Congruent triangles</strong> are identical in size and shape. The criteria for proving two triangles congruent are SSS (all three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), AAS (two angles and a non-included side equal), and HL (hypotenuse-leg, for right triangles only).</p>

<p><strong>Similar triangles</strong> have the same shape but not necessarily the same size, their corresponding angles are equal and their corresponding sides are proportional. The criteria for similarity are AA (two angles equal, since the third is automatically equal too), SAS similarity (two sides proportional with the included angle equal), and SSS similarity (all three sides proportional).</p>

<p><strong>Worked Example:</strong> Triangle ABC has sides 6, 8, and 10. Triangle DEF has sides 9, 12, and 15. Are the triangles similar, and what is the scale factor from ABC to DEF?</p>

<p>All three ratios are equal, so the triangles are similar by SSS similarity, with a scale factor of 1.5.</p>

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<p class="prompt">1. Which congruence criterion proves two triangles congruent using two sides and the included angle between them?</p>

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<p><span class="label">Explanation:</span> SAS (side-angle-side) uses two sides and the angle between them.</p>

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<p class="prompt">2. Which similarity criterion is sufficient to prove two triangles similar using only their angles?</p>

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<p><span class="label">Explanation:</span> AA (angle-angle) similarity works because if two angles of one triangle equal two angles of another, the third pair must be equal too, since angles in a triangle sum to 180°.</p>

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<p class="prompt">3. Triangle ABC has sides 6, 8, and 10. Triangle DEF has sides 9, 12, and 15. Are the triangles similar, and what is the scale factor from ABC to DEF?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-3',true)">A) Similar, scale factor 1.5</button>

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<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">C) Not similar</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">D) Congruent, scale factor 1</button>

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<p><span class="label">Explanation:</span> This is the worked example above: all three side ratios equal 1.5, so the triangles are similar with that scale factor.</p>

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<p class="prompt">4. Two triangles are similar with a scale factor of 3. If the smaller triangle has a perimeter of 20, what is the perimeter of the larger triangle?</p>

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<p><span class="label">Explanation:</span> Perimeter is a linear measurement, so it scales by the same factor as the sides: 20 × 3 = 60.</p>

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<p class="prompt">5. In similar triangles with a scale factor of k, how does the ratio of their areas compare?</p>

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<p><span class="label">Explanation:</span> Since area involves two linear dimensions multiplied together, the area ratio is the square of the linear scale factor, k².</p>

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<p class="prompt">6. Triangle PQR ~ Triangle STU. If PQ = 12, ST = 8, and QR = 15, what is TU?</p>

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<p><span class="label">Explanation:</span> The scale factor from PQR to STU is ST ÷ PQ = 8 ÷ 12 = <span style="font-family:'Cambria Math',Georgia,serif;">2/3</span>. So TU = QR × <span style="font-family:'Cambria Math',Georgia,serif;">2/3</span> = 15 × <span style="font-family:'Cambria Math',Georgia,serif;">2/3</span> = 10.</p>

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<h2 id="inequality">4. The Triangle Inequality</h2>

<p>The <strong>triangle inequality theorem</strong> states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this isn't true for a set of three lengths, no triangle can be formed.</p>

<p><strong>Worked Example:</strong> Can a triangle have side lengths 5, 7, and 13?</p>

<p>Since 12 is less than 13, the two shorter sides can't reach far enough to close the triangle. No, this is not a valid triangle.</p>

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<p class="prompt">1. Can a triangle have side lengths 4, 6, and 9?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-1',false)">C) Only if it's a right triangle</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-1',false)">D) Cannot be determined</button>

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<p><span class="label">Explanation:</span> Check the two shorter sides: 4 + 6 = 10, and 10 > 9, so yes, this is a valid triangle.</p>

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<p class="prompt">2. Can a triangle have side lengths 3, 4, and 8?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">C) Only if isosceles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">D) Cannot be determined</button>

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<p><span class="label">Explanation:</span> 3 + 4 = 7, and 7 < 8, so these three lengths cannot form a triangle.</p>

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<p class="prompt">3. A triangle has sides of length 7 and 12. Which of the following could be the length of the third side?</p>

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<p><span class="label">Explanation:</span> The third side must be greater than 12 − 7 = 5 and less than 12 + 7 = 19. Only 15 falls strictly between 5 and 19.</p>

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<p class="prompt">4. A triangle has two sides of length 9 each. What is the range of possible lengths for the third side?</p>

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<p><span class="label">Explanation:</span> The third side must be greater than 9 − 9 = 0 and less than 9 + 9 = 18, giving 0 < x < 18.</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° (think "corner"), supplementary sums to 180° (think "straight line"). Confusing the two flips your entire answer.</li>

<li><strong>Assuming any two angles formed by a transversal are automatically equal.</strong> Only specific pairs, like corresponding, alternate interior, and alternate exterior, are equal. Co-interior (same-side interior) angles are supplementary, not equal.</li>

<li><strong>Using SSA as a congruence criterion.</strong> Side-Side-Angle, with the angle not included between the two given sides, does not reliably prove triangles congruent, it's a common trap since it looks similar to the valid SAS.</li>

<li><strong>Applying the linear scale factor to area or the area scale factor to length.</strong> Remember: length scales by k, area scales by k², and volume scales by k³. Mixing these up is one of the most common similarity errors.</li>

<li><strong>Only checking one pair of sides for the triangle inequality.</strong> To confirm three lengths form a valid triangle, technically all three combinations should satisfy the inequality, but checking that the two shortest sides sum to more than the longest side is always sufficient on its own.</li>

</ul>

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

<h3>Do I need to memorize every congruence and similarity criterion by name?</h3>

<p>Recognizing which pieces of information (sides, angles) are given, and whether that combination is enough to guarantee congruence or similarity, matters more than reciting the abbreviations. Still, knowing the names, SSS, SAS, ASA, AAS, HL, AA, helps you communicate and check your reasoning quickly.</p>

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<h3>What's the difference between congruent and similar triangles?</h3>

<p>Congruent triangles are exactly the same size and shape, every corresponding side and angle matches exactly. Similar triangles have the same shape and equal corresponding angles, but their sides are only proportional, not necessarily equal, so one triangle is a scaled version of the other.</p>

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<h3>Why does the triangle inequality theorem matter on the SAT?</h3>

<p>It's a quick way to rule out impossible triangles without needing any angle information. Questions sometimes give you three side lengths and ask whether a triangle can be formed, or ask for the possible range of an unknown side, both of which rely directly on this theorem.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/sat-lines-angles-triangles-quiz-1">Lines, Angles, and Triangles quizzes</a> in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/sat-lines-angles-triangles-quiz-1">Practice Lines, Angles & Triangles Free</a>

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

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