Geometry Theorems About Triangles Worksheet (PDF)

Geometry math worksheet: Theorems About Triangles – Maths PDF

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

These 20 problems practice angle sum, exterior angle, isosceles, midsegment, centroid and triangle inequality theorems (Common Core G-CO.C.10). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with gardens, roof trusses, paths and coordinates.

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

Triangle theorems worksheet: exercises 1 to 10

Exercise 1 – Third angle

Two angles of a triangle measure \(52^{\circ}\) and \(67^{\circ}\). Find the third angle.

Exercise 2 – Right triangle

One acute angle of a right triangle is \(38^{\circ}\). Find the other acute angle.

Exercise 3 – Equilateral triangle

In an equilateral triangle each side is \(3x – 6\) and the perimeter is \(54\). Find \(x\) and each angle measure.

Exercise 4 – Exterior angle

Exterior angle at B.

Find the exterior angle at B.

Exercise 5 – Remote angle

An exterior angle of a triangle measures \(140^{\circ}\). One remote interior angle is \(85^{\circ}\). Find the other remote interior angle.

Exercise 6 – Isosceles apex

Isosceles triangle ABC.

In isosceles triangle ABC, AB = AC and \(m\angle A = 40^{\circ}\). Find the base angles.

Exercise 7 – Isosceles base

A base angle of an isosceles triangle measures \(54^{\circ}\). Find the vertex angle.

Exercise 8 – Angles in a ratio of 1 : 2 : 3

The angles of a triangle measure \(x\), \(2x\) and \(3x\) degrees. Find each angle.

Exercise 9 – Angles with x

The angles of a triangle measure \((2x + 10)^{\circ}\), \((3x – 5)^{\circ}\) and \((x + 25)^{\circ}\). Find x and each angle.

Exercise 10 – Exterior angle with x

The remote interior angles of a triangle measure \((2x + 11)^{\circ}\) and \((3x + 13)^{\circ}\), and the exterior angle is \((7x – 16)^{\circ}\). Find x and the exterior angle.

Exercises 11 and up

Exercise 11 – Midsegment length

D and E are midpoints.

D and E are the midpoints of AB and AC. If BC = 18, find DE.

Exercise 12 – Midsegment with x

The midsegment DE of triangle ABC has \(DE = 2x + 3\) and the parallel side has \(BC = 5x – 6\). Find x, DE and BC.

Exercise 13 – Midsegment on the plane

Triangle ABC and midsegment MN.

Triangle ABC has A(0, 0), B(8, 2) and C(2, 8). M and N are the midpoints of AB and AC. Find the coordinates of M and N and show that MN is half of BC.

Exercise 14 – Centroid and a median

Medians of triangle ABC.

The median from A has length 15. The centroid G lies on it. Find AG and the distance from G to the midpoint of the opposite side.

Exercise 15 – Centroid coordinates

Find the centroid of the triangle with vertices (0, 0), (6, 0) and (3, 9).

Exercise 16 – Can it be a triangle?

Can a triangle have sides of 5, 7 and 13? Explain.

Exercise 17 – Third side range

Two sides of a triangle measure 9 and 14. Find all possible lengths of the third side.

Exercise 18 – Garden angles

The angles of a triangular garden are in the ratio \(2 : 3 : 5\). Find the three angles and say what kind of triangle it is.

Exercise 19 – Roof truss

Roof truss.

An isosceles roof truss has a 110° angle at the peak. Find the angle the rafters make with the horizontal beam.

Exercise 20 – Walking paths

Maya walks 120 feet east and then 200 feet along a straight path to a pond. A third straight path joins the pond back to her starting point. Can that third path be 70 feet long? Give the possible lengths.

Key facts behind this triangle theorems worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the triangle theorems worksheet as a PDF with the button below and keep it with your notes.

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