Angle Sum Property of a Triangle
Let's do a simple experiment to find out the angle sum property of a triangle. Can you guess what it is?
Do this: Add a triangle to the panel.
Add another triangle and rotate it 180°(invert it).
Drag the second triangle and make it attach to the first triangle without changing its rotation.
Add another triangle and drag it to attach it to the second triangle.
The three angles are said to constitute one single angle. As it can be sees, this angle is a straight angle and thus, has a measure of 180°.
Drag and group the three triangles and move them or rotate them. What do you say? They for a
Another Activity: Take a piece of paper and cut out a triangle, say, ∆ABC. Make the altitude AM by folding ∆ABC such that it passes through A. Fold the paper triangle such that each vertex/corners A, B and C touches the point M.
Another Activity: Draw any number of triangles, say namely ∆ABC, ∆PQR and ∆XYZ. Now, use a protractor and measure each of the angles of all these triangles individually.
| Name of ∆ | Measures of Angles | Sum of the Measures of the three Angles |
|---|---|---|
| ∆ABC | m∠A + m∠B + m∠C = | |
| ∆PQR | m∠P + m∠Q + m∠R = | |
| ∆XYZ | m∠X + m∠Y + m∠Z = |
Allowing some amount of marginal measurement errors, we find that the sum measure of all the three angles for each triangle (last column) always gives 180° (or nearly 180°).
This statement can also be proved using the exterior angle of a triangle property.
- By exterior angle property
- adding
∠ c to both the sides - But ∠c and ∠d form a linear pair so it is 180°.
- RHS =
° and Hence, ∠a + ∠b + ∠c = °.
In the given figure Fig find m∠P.
Fig
Solution:
By angle sum property of a triangle,
m∠P + 47° + 52° =
Therefore
m∠P = 180° –
= 180° – 99° =
1. Two angles of a triangle are 30° and 80°. Find the third angle.
Solution:
Given:
Two angles of the triangle are
To find: The measure of the third angle.
Sum of Interior Angles:
The sum of the interior angles of a triangle is
Angle 1 + Angle 2 + Angle 3 =
Substitute the Known Angles:
Given Angle 1 =
30° + 80° + Angle 3 =
Solve for the Third Angle:
Combine the known angles:
Subtract 110° from both sides:
Angle 3 = 180° -
Angle 3 =
Therefore,The measure of the third angle is 70°.
2. One of the angles of a triangle is 80° and the other two angles are equal. Find the measure of each of the equal angles.
Solution:
Given:
One angle of the triangle is
The other two angles are equal.
To find: The measure of each of the equal angles.
Sum of Interior Angles:
The sum of the interior angles of a triangle is
Angle 1 + Angle 2 + Angle 3 =
Given Information:
Let the equal angles be Angle 1 = Angle 2 = x.
Given Angle 3 =
Set Up the Equation:
Substitute the known values into the sum of angles equation:
x + x +
Combine like terms:
Solve for x:
Subtract 80° from both sides:
2x = 180° -
2x =
Divide both sides by 2:
x =
x =
Therefore,The measure of each of the equal angles is 50°.
3. The three angles of a triangle are in the ratio 1
1. Find all the angles of the triangle.Classify the triangle in two different ways.
Solution:
Given:
The three angles of a triangle are in the ratio
To find: The measure of each angle.
Classify the triangle in two different ways.
Express the Angles in Terms of a Variable:
Let the angles be x, 2x, and x based on the given ratio 11.
Sum of Interior Angles:
The sum of the interior angles of a triangle is 180°.
x + 2x + x =
Combine Like Terms:
Simplify the equation:
Divide both sides by 4: x =
x =
Substitute x = 45°:
First angle = x =
Second angle = 2x = 2 × 45° =
Third angle = x =
Therefore,The three angles are
Classification of the Triangle:
1.By Angles:
Since one of the angles is
2.By Sides:
Since two of the angles are equal (45° each), the triangle is an
isosceles triangle
Therefore, the triangle is classified as both a right-angled triangle and an isosceles triangle.
Property: The total sum of the measures of all the three angles of a triangle is 180°.