Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Early problems apply one theorem. Later problems combine algebra, coordinates and short proofs.
Standards: G-CO.C.10, G-C.A.3 · Level: Grade 10 Geometry
Problem 1 Third angle of a triangle ★★★
Two angles of a triangle measure 47° and 68°. Find the third angle.
Problem 2 Exterior angle ★★★
In △ABC, m∠A = 52° and m∠B = 79°. Side BC is extended to D. Find m∠ACD.
Problem 3 Base angles ★★★
The vertex angle of an isosceles triangle measures 38°. Find each base angle.
Problem 4 Converse of the isosceles theorem ★★★
In △ABC, ∠B ≅ ∠C, AB = 3x − 4 and AC = x + 10. Find x and AB.
Problem 5 Midsegment length ★★★
M and N are the midpoints of AB and AC. If BC = 26.4 centimeters, find MN.
Problem 6 Midsegment with algebra ★★★
In △ABC, MN joins the midpoints of AB and AC. MN = 2x + 7 and BC = 8x − 10. Find x, MN and BC.
Problem 7 Angle sum with algebra ★★★
The angles of a triangle measure (2x + 5)°, (3x − 10)° and (x + 35)°. Find x and all three angles.
Problem 8 Exterior angle with algebra ★★★
An exterior angle of a triangle measures (8x − 9)°. The remote interior angles measure (3x + 14)° and (2x + 25)°. Find x and the exterior angle.
Problem 9 Parts of a median ★★★
The centroid G lies on median AD, and AD = 27. Find AG and GD.
Problem 10 Centroid ratio with algebra ★★★
The medians of a triangle meet at G. On one median, AG = 3x − 2 and GD = x + 4. Find x, AG and the length of the whole median.
Problem 11 Centroid coordinates ★★★
Triangle ABC has A(1, 2), B(7, −1) and C(4, 8). Find the centroid.
Problem 12 Circumcenter of a right triangle ★★★
Triangle PQR has P(0, 0), Q(10, 0) and R(0, 24). Find the circumcenter and the radius of the circumscribed circle.
Problem 13 Inscribed circle radius ★★★
A right triangle has legs of 9 and 12 and a hypotenuse of 15. Find the radius of its inscribed circle using area = radius × semiperimeter.
Problem 14 Name the point of concurrency ★★★
Name the point of concurrency described by each statement.
- The point where the three perpendicular bisectors meet.
- The point where the three angle bisectors meet.
- The point where the three medians meet.
- The point where the three altitudes meet.
Problem 15 Fire station location ★★★
Three towns sit at A(0, 0), B(12, 0) and C(0, 16), with each unit equal to one mile. A county plans a fire station the same distance from all three towns. Find its location and that distance.
Problem 16 Balance point of a plate ★★★
A robotics club cuts a triangular aluminum plate with vertices (0, 0), (9, 0) and (3, 12), measured in centimeters. Where should a single support rod go to balance the plate?
Problem 17 Perimeter from midsegments ★★★
The sides of △ABC are AB = 18, BC = 26 and CA = 22. The midpoints of the three sides form a smaller triangle. Find its three side lengths and its perimeter, and compare with the perimeter of △ABC.
Problem 18 Prove the exterior angle theorem ★★★
In △ABC, side BC is extended to D. Prove that m∠ACD = m∠A + m∠B. Write a two-column proof.
Problem 19 Prove the isosceles triangle theorem ★★★
Given: AB ≅ AC in △ABC. Prove: ∠B ≅ ∠C. Draw the bisector of ∠A meeting BC at D, then write a two-column proof.
Problem 20 Midsegment on the coordinate plane ★★★
Triangle ABC has A(2, 10), B(−4, 2) and C(10, 0). Let M and N be the midpoints of AB and AC. Show that MN is parallel to BC and half as long.
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