Lesson HOW TO solve problems on the angles of triangles - Examples

How to solve problems on the angles of triangles - Examples

In this lesson you will find the solutions of the typical problems on the angles of triangles.
To solve this kind of problems we use the basic property that the sum of the angles of a triangle is equal to 180°.
See the lesson Sum of the interior angles of a triangle under the topic Triangles of the section Geometry in this site.


Problem 1
Find the third angle in the triangle, if the first angle is 35° and the second angle is 73°.

Solution
Since the sum of the angles of a triangle is equal to 180°, the third angle is
180°- (35° + 73°) = 180° - 108° = 72°.

Answer. The third angle of the triangle is 72°.


Problem 2
In a triangle, the first angle is 35° and the second angle is in 23° greater.
Find the third angle in the triangle.

Solution
The second angle is equal to 35° + 23° = 58° in accordance with the problem condition.
Since the sum of the angles of a triangle is equal to 180°, the third angle is equal to
180°- (35° + 58°) = 180° - 93° = 87°.

Answer. The third angle of the triangle is 87°.


Problem 3
In a triangle, the second angle is in 35° greater than the first one and the third angle is in 14° than the second one.
Find the angles of the triangle.

Solution
Let x be the angle measure of the first angle (in degrees).
Then the second angle is x + 35 degrees, and the third angle is (x + 35) + 14 degrees in accordance with the problem condition.
Since the sum of the angles of a triangle is equal to 180°, you can write the equation
x + (x +35) + ((x + 35) + 14) = 180.
Simplify this equation step by step and solve it:
x + x + 35 + x + 35 + 14 = 180,
3x + 84 = 180,
3x = 180 - 84,
3x = 96,
x = 32.

So, the first angle of the triangle has the angle measure of 32°.
Then the second angle is 32° + 35° = 67°, and the third angle is 67° + 14° = 81° in accordance with the problem condition.
You can check that the sum of the angles is 180°: 32° + 67° + 81° = 180°.

Answer. The angles of the triangle are 32°, 67° and 81°.


Problem 4
Find the angles of the triangle ABC, if the angle LB is twice the angle LA and the angle LC is three times the angle LA.

Solution
Let x be the angle measure of the angle A (in degrees).
Then the angle LB is 2x and the angle LC is 3x in accordance with the problem condition.
Since the sum of the angles of a triangle is equal to 180°, you can write the equation
x + 2x + 3x = 180.
Simplify this equation step by step and solve it:
6x = 180,
x = 30.

So, the angle LA of the triangle is 30°.
Then the angle LB is 2*30° = 60°, and the angle LC is 3*30° = 90° in accordance with the problem condition.
You can check that the sum of the angles is 180°: 30° + 60° + 90° = 180°.

Answer. The angles of the triangle are 30°, 60° and 90°. This is the remarkable right angle triangle.


Problem 5
Prove that the sum of the interior angles of any convex quadrilateral is equal to 360°.

Solution
The proof is very simple.
In the given quadrilateral, draw the straight line through any two opposite vertices of the quadrilateral, which is the diagonal of the quadrilateral.
It divides the quadrilateral in two triangles that lies in different half-planes relative to the straight line.
The sum of the angles is equal to 180° for each of the two triangles.
The sum of the angles of the quadrilateral is equal to the sum of the angles of these two triangles, that is 2*180° = 360°.
The statement is proved.