Geometry | Free ACT Math Course | The School of Mathematics

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<h1>ACT Math: Geometry</h1>

Geometry is the broadest Plane Geometry topic on the ACT, and this free, complete lesson covers five core areas: triangle types and the Pythagorean theorem, special right triangles (45-45-90 and 30-60-90), triangle congruence, triangle similarity, and quadrilateral properties. 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-geometry-quiz-1">Practice Geometry 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="#triangle-types">Triangle Types and the Pythagorean Theorem</a></li>

<li><a href="#special-triangles">Special Right Triangles</a></li>

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

<li><a href="#similarity">Triangle Similarity</a></li>

<li><a href="#quadrilaterals">Quadrilateral Properties</a></li>

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

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

</ol>

</nav>

<h2 id="triangle-types">1. Triangle Types and the Pythagorean Theorem</h2>

<p>Triangles are classified by their sides (scalene: no equal sides; isosceles: two equal sides; equilateral: all sides equal) and by their angles (acute, right, obtuse). For any right triangle with legs a and b and hypotenuse c:</p>

<p><strong>Worked Example:</strong> A right triangle has legs 9 and 12. Find the hypotenuse.</p>

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<p class="prompt">1. A right triangle has legs 5 and 12. What is the hypotenuse?</p>

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<p><span class="label">Explanation:</span> 5² + 12² = 25 + 144 = 169, and √169 = 13.</p>

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<p class="prompt">2. A right triangle has a hypotenuse of 25 and one leg of 7. What is the other leg?</p>

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<p><span class="label">Explanation:</span> 25² − 7² = 625 − 49 = 576, and √576 = 24.</p>

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<p class="prompt">3. Which set of side lengths could NOT form a right triangle?</p>

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<p><span class="label">Explanation:</span> For 7, 10, 13: 7² + 10² = 149, but 13² = 169. Since these aren't equal, this is not a right triangle. The other three are all valid Pythagorean triples.</p>

</div>

<p class="prompt">4. An isosceles triangle has two sides measuring 10 and a third side measuring 16. Does this satisfy the triangle inequality (the two shorter sides must sum to more than the longest side)?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-4',true)">A) Yes, since 10 + 10 > 16</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">B) No, since 10 + 10 < 16</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">C) Cannot be determined</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">D) Only if it's also a right triangle</button>

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<p><span class="label">Explanation:</span> 10 + 10 = 20, and 20 > 16, so this is a valid triangle.</p>

</div>

<p class="prompt">5. Each angle of an equilateral triangle measures how many degrees?</p>

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<p><span class="label">Explanation:</span> All three angles are equal, and they sum to 180°, so each measures 180° ÷ 3 = 60°.</p>

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<h2 id="special-triangles">2. Special Right Triangles</h2>

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<p class="figure-caption">45-45-90: legs equal, hypotenuse = x√2</p>

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<p class="figure-caption">30-60-90: sides x, x√3, 2x</p>

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<p><strong>Worked Example:</strong> A 45-45-90 triangle has a leg of length 8. Find the hypotenuse.</p>

<p class="step-math">hypotenuse = 8√2</p>

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<p class="prompt">1. A 45-45-90 triangle has a leg of length 6. What is the hypotenuse?</p>

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<p><span class="label">Explanation:</span> The hypotenuse is the leg times √2: 6√2.</p>

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<p class="prompt">2. A 45-45-90 triangle has a hypotenuse of 10√2. What is the length of each leg?</p>

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<p><span class="label">Explanation:</span> Divide the hypotenuse by √2: 10√2 ÷ √2 = 10.</p>

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<p class="prompt">3. A 30-60-90 triangle has a short leg (opposite the 30° angle) of length 5. What is the hypotenuse?</p>

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<p><span class="label">Explanation:</span> The hypotenuse is always twice the short leg: 2(5) = 10.</p>

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<p class="prompt">4. A 30-60-90 triangle has a short leg of length 5. What is the length of the long leg?</p>

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<p><span class="label">Explanation:</span> The long leg is always the short leg times √3: 5√3.</p>

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<p class="prompt">5. A 30-60-90 triangle has a hypotenuse of length 14. What is the short leg?</p>

</div>

<p><span class="label">Explanation:</span> The short leg is half the hypotenuse: 14 ÷ 2 = 7.</p>

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

<p><strong>Congruent triangles</strong> are identical in size and shape. Valid congruence criteria include SSS (all three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse-leg, for right triangles only). SSA (side-side-angle, with the angle not between the two sides) is <strong>not</strong> a valid criterion.</p>

<p class="prompt">1. Which congruence criterion uses all three sides?</p>

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<p><span class="label">Explanation:</span> SSS (side-side-side) proves congruence using all three side lengths.</p>

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<p class="prompt">2. Which congruence criterion uses two angles and the included side?</p>

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

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<p class="prompt">3. Which congruence criterion applies specifically to right triangles, comparing the hypotenuse and one leg?</p>

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<p><span class="label">Explanation:</span> HL (hypotenuse-leg) is a right-triangle-specific shortcut, comparing the hypotenuse and one leg.</p>

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<p class="prompt">4. Two triangles have all three pairs of corresponding sides equal in length. What criterion proves them congruent?</p>

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<p><span class="label">Explanation:</span> Three equal corresponding sides guarantee congruence by SSS.</p>

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<p class="prompt">5. Is SSA (side-side-angle, with the angle not between the two given sides) a valid congruence criterion?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-5',true)">A) No, SSA does not reliably prove triangles congruent</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">B) Yes, it's equivalent to SAS</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">C) Yes, it's equivalent to ASA</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">D) Yes, but only for right triangles</button>

</div>

<p><span class="label">Explanation:</span> SSA is a well-known trap, it can produce two different possible triangles from the same information, so it does not reliably prove congruence.</p>

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<h2 id="similarity">4. Triangle Similarity</h2>

<p><strong>Similar triangles</strong> have the same shape but not necessarily the same size, corresponding angles are equal and corresponding sides are proportional. Valid criteria: AA (two angles equal), 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 they similar? What's the scale factor?</p>

<p>9 ÷ 6 = 1.5, 12 ÷ 8 = 1.5, 15 ÷ 10 = 1.5. All three ratios match, so the triangles are similar with a scale factor of 1.5.</p>

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<p class="prompt">1. Which similarity criterion uses only angle measures?</p>

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<p><span class="label">Explanation:</span> AA (angle-angle) similarity works because if two angles match, the third automatically matches too, since triangle angles sum to 180°.</p>

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<p class="prompt">2. Two triangles are similar with a scale factor of 3. If the smaller triangle has a perimeter of 20, what is the larger triangle's perimeter?</p>

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<p><span class="label">Explanation:</span> Perimeter scales linearly with the sides: 20 × 3 = 60.</p>

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<p class="prompt">3. In similar triangles with a scale factor of k, how does the area ratio compare?</p>

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

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<p class="prompt">4. 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 class="frac"><span class="num">2</span><span class="den">3</span></span>. So TU = QR × <span class="frac"><span class="num">2</span><span class="den">3</span></span> = 15 × <span class="frac"><span class="num">2</span><span class="den">3</span></span> = 10.</p>

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<h2 id="quadrilaterals">5. Quadrilateral Properties</h2>

<tr><th>Quadrilateral</th><th>Key properties</th></tr>

<tr><td>Parallelogram</td><td>Opposite sides parallel and equal; opposite angles equal; consecutive angles supplementary</td></tr>

<tr><td>Rectangle</td><td>A parallelogram with four right angles</td></tr>

<tr><td>Rhombus</td><td>A parallelogram with four equal sides; diagonals are perpendicular</td></tr>

<tr><td>Square</td><td>A rectangle and a rhombus at once: equal sides and right angles</td></tr>

<tr><td>Isosceles trapezoid</td><td>One pair of parallel sides; base angles are congruent</td></tr>

</table>

<p class="prompt">1. In a parallelogram, consecutive (adjacent) angles are always what?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pe-1',true)">A) Supplementary</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-1',false)">B) Equal</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-1',false)">C) Complementary</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-1',false)">D) Right angles</button>

</div>

<p><span class="label">Explanation:</span> Consecutive angles in a parallelogram always sum to 180°, the same relationship as co-interior angles along a pair of parallel sides.</p>

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<p class="prompt">2. Which quadrilateral has all four sides equal AND all four angles equal to 90°?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pe-2',true)">A) Square</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-2',false)">B) Rhombus</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-2',false)">C) Rectangle</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-2',false)">D) Trapezoid</button>

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<p><span class="label">Explanation:</span> A square is both a rhombus (equal sides) and a rectangle (right angles) at the same time.</p>

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<p class="prompt">3. A rhombus's diagonals are always what, relative to each other?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pe-3',true)">A) Perpendicular</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-3',false)">B) Equal in length</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-3',false)">C) Parallel</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-3',false)">D) Congruent to the sides</button>

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<p><span class="label">Explanation:</span> A rhombus's diagonals always intersect at right angles, though they aren't generally equal in length (that property belongs to rectangles).</p>

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<p class="prompt">4. In an isosceles trapezoid, what is true about the base angles?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pe-4',true)">A) Each pair of base angles is congruent</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-4',false)">B) All four angles are equal</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-4',false)">C) Opposite angles are supplementary</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-4',false)">D) The diagonals are perpendicular</button>

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<p><span class="label">Explanation:</span> An isosceles trapezoid has one pair of parallel sides, and the two angles along each parallel side (the base angles) are congruent to each other.</p>

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<p class="prompt">5. A parallelogram has one angle measuring 65°. What is the measure of the angle consecutive (adjacent) to it?</p>

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<p><span class="label">Explanation:</span> Consecutive angles in a parallelogram are supplementary: 180° − 65° = 115°.</p>

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

<li><strong>Using SSA as a congruence criterion.</strong> It's one of the most common traps, since it resembles the valid SAS but doesn't reliably guarantee a unique triangle.</li>

<li><strong>Mixing up the 45-45-90 and 30-60-90 side ratios.</strong> A 45-45-90 triangle has two equal legs with hypotenuse leg × √2. A 30-60-90 triangle has three distinct sides in the ratio x : x√3 : 2x.</li>

<li><strong>Applying the linear scale factor to area instead of squaring it.</strong> Length scales by k, but area scales by k², mixing these up is one of the most common similarity errors.</li>

<li><strong>Assuming all parallelograms are rectangles, or all rhombi are squares.</strong> A rectangle and a rhombus are each special cases of a parallelogram, with additional properties (right angles, or equal sides) that a general parallelogram doesn't automatically have.</li>

<li><strong>Confusing congruence and similarity.</strong> Congruent figures are identical in both shape and size; similar figures share the same shape but can differ in size, having proportional rather than equal sides.</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 are given, and whether that combination guarantees congruence or similarity, matters more than reciting the abbreviations. Still, knowing the names helps you communicate your reasoning quickly and avoid the SSA trap.</p>

</div>

<h3>Why are the 45-45-90 and 30-60-90 triangles worth memorizing directly?</h3>

<p>These ratios come from cutting a square along its diagonal (45-45-90) or an equilateral triangle in half (30-60-90). Knowing the exact ratios by heart lets you skip the Pythagorean theorem entirely whenever one of these specific triangles appears.</p>

</div>

<h3>What's the fastest way to remember quadrilateral properties?</h3>

<p>Think in terms of a hierarchy: a parallelogram is the base case (parallel and equal opposite sides). A rectangle adds right angles. A rhombus adds equal sides. A square is both at once. Trapezoids are the outlier, needing only one pair of parallel sides.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/act-geometry-quiz-1">Geometry 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-geometry-quiz-1">Practice Geometry 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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