Geometry Theorems About Triangles Test Solutions and Scoring (PDF)

Geometry math test solutions: Theorems About Triangles – Maths PDF

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This triangle theorems test answers for Geometry covers theorems about triangles (Common Core G-CO.C.10), with a free PDF to print.

Use this answer key and scoring guide for the triangle theorems test. Each solution shows the full work, and the scoring note tells how the points are split between method and final answer. Partial credit is listed so that grading stays consistent from one paper to the next.

Geometry · Common Core State Standards: G-CO.C.10

Triangle theorems test answers: worked solutions

Question 1 – Angle sum [2 pts]

Angles 32°, 52°, 96°.

\(x + (x + 20) + 3x = 180\), so \(5x = 160\) and \(x = 32\).

Angles: 32°, 52° and 96°. One angle is greater than 90°, so the triangle is obtuse.

32°, 52°, 96°; obtuse

Scoring: 1 point for the equation and x, 1 point for the three angles and the classification.

Question 2 – Exterior angle [2 pts]

Remote interior angles: 55° and 60°.

\(y = 55^{\circ} + 60^{\circ} = 115^{\circ}\)

115°

Scoring: 1 point for using the theorem, 1 point for the value.

Question 3 – Apex angle [2 pts]

The three angles add to 180°: \((6x + 4) + 2(2x + 8) = 180\).

\(10x + 20 = 180\), so \(x = 16\).

Apex angle: \(6(16) + 4 = 100^{\circ}\) (base angles 40°).

x = 16; apex angle 100°

Scoring: 1 point for the equation, 1 point for x and the apex angle.

Question 4 – Midsegment [3 pts]

DE = 5 is half of BC = 10.

\(D(1, 4)\) and \(E(6, 4)\)
\(DE = 5\) and \(BC = 10\)
DE is parallel to BC (both horizontal) and half as long.

Scoring: 1 point per part.

Question 5 – Centroid [2 pts]

\(\left( \frac{1 + 7 + 4}{3}, \frac{2 + 2 + 11}{3} \right) = (4, 5)\).

The centroid is two thirds along the median: \(\frac{2}{3} \cdot 12 = 8\).

(4, 5); 8 units

Scoring: 1 point for the centroid, 1 point for the distance.

Question 6 – Triangle inequality [3 pts]

\(8 + 9 = 17\) is not greater than 17, so no.
\(15 – 6 = 9 < x < 21 = 15 + 6\)
The third side can be at most 20: \(6 + 15 + 20 = 41\).

(a) No; (b) 9 < x < 21; (c) 41

Scoring: 1 point per part.

Question 7 – Why 180 [2 pts]

The parallel line creates alternate interior angles at B and C that are congruent to angles B and C of the triangle.

Those two angles and angle A fill a straight line at A, which measures 180°.

Scoring: 1 point for alternate interior angles, 1 point for the straight line.

Question 8 – Ramp angles [2 pts]

Total parts: 12, so one part is \(180^{\circ} \div 12 = 15^{\circ}\).

Angles: 45°, 60°, 75°. No angle is 90°, so it is not a right triangle.

45°, 60°, 75°; not right

Scoring: 1 point for the angles, 1 point for the conclusion.

Question 9 – Equilateral side [2 pts]

All sides are equal: \(4x + 2 = 6x – 8\), so \(x = 5\).

Check: \(3(5) + 7 = 22\), and \(4(5) + 2 = 22\). Each side is 22 and each angle is \(60^{\circ}\).

x = 5; side 22; angles 60°

Scoring: 1 point for x and the side, 1 point for the angles.

Total score

Total

20 points

Key facts behind this triangle theorems test answers

Key facts

Triangle angle sum: 180 degrees.
Exterior angle = sum of the two remote interior angles.
Isosceles triangle: base angles are congruent; equilateral angles are 60 degrees each.
Midsegment: parallel to the third side and half its length.
Medians meet at the centroid, two thirds of the way from each vertex.

How to use this triangle theorems test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Geometry chapters: Geometry math.

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