Use angle relationships, triangle rules, similarity, and the Pythagorean theorem to find missing measures.
You find missing angles and side lengths using geometric relationships. That includes angles on a line and around a point, vertical angles, angles formed when parallel lines are cut by a transversal, the triangle angle sum, isosceles triangle properties, similar triangles and their proportional sides, and the Pythagorean theorem. Many questions describe the figure in words with no picture, so part of the work is drawing it yourself and labeling what you are told.
Geometry and Trigonometry is about 15 percent of the 44 math questions, and lines, angles, and triangles are usually around 2 to 3 of them.
Written for this skill, with the answer explained. 149 more are in the app.
In triangle ABC shown, what is the value of x?
The sum of interior angles of a pentagon is:
Why: A polygon with n sides has interior angles summing to (n - 2)(180), so a pentagon gives (5 - 2)(180) = 540. Answering 360 uses (5 - 3) or the sum for a quadrilateral. Answering 720 is the sum for a hexagon, and 480 matches no polygon, since the sums run 180, 360, 540, 720.
Lines m and n are parallel and are crossed by transversal t. Two same-side exterior angles formed have measures (5x - 4) degrees and (3x + 8) degrees. What is the value of x?
Why: Same-side exterior angles formed by a transversal across parallel lines are supplementary, so (5x - 4) + (3x + 8) = 180. That gives 8x + 4 = 180, so 8x = 176 and x = 22. Answering 6 comes from setting the two expressions equal, which would hold only if the angles were congruent. Answering 24 makes the angles 116 and 80, a total of 196, and 11 makes them 51 and 41, a total of 92.
In the figure, line ℓ and line m are parallel, and line t intersects line ℓ at P and line m at Q. The measure of angle BPS is 47°. What is the measure of angle DQT, in degrees?
Why: Angle BPS corresponds to angle DQS, so angle DQS is 47°. Angle DQT and angle DQS lie along line t on opposite sides of Q, so they are supplementary: 180 - 47 = 133. Choosing 47 stops at the corresponding angle. Answering 43 means you took the complement of the given angle. Answering 47 means you gave the corresponding angle instead of its supplement. Answering 94 means you doubled the given angle.
Two triangles are similar if two pairs of their angles are equal, which forces the third pair to match as well. Parallel lines and shared angles are the usual sources. Once similar, corresponding sides are in the same ratio.
Corresponding angles are equal, alternate interior angles are equal, and vertical angles are equal. Same-side interior angles are supplementary, meaning they add to 180 degrees. Every angle in the figure is either equal to a given angle or supplementary to it.
Knowing the 30-60-90 and 45-45-90 side ratios saves time, and the 3-4-5 and 5-12-13 patterns help you spot answers fast. The reference sheet at the start of the math section does list the special right triangle ratios, so recognition matters more than memorization.
Questions on lines, angles, and triangles alone, with explanations on every one, and the app tracks which ones you keep missing.
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