What if you were presented with an isoceles triangle and told that its base angles measure
Watch the first part of this video.
An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides of the isosceles triangle are called the legs. The other side is called the base and the angles between the base and the congruent sides are called base angles. The angle made by the two legs of the isosceles triangle is called the vertex angle.
Investigation: Isosceles Triangle Construction
Tools Needed: pencil, paper, compass, ruler, protractor
- Using your compass and ruler, draw an isosceles triangle with sides of 3 in, 5 in and 5 in. Draw the 3 in side (the base) horizontally 6 inches from the top of the page.
- Now that you have an isosceles triangle, use your protractor to measure the base angles and the vertex angle.
The base angles should each be
We can generalize this investigation into the Base Angles Theorem.
Base Angles Theorem: The base angles of an isosceles triangle are congruent.
To prove the Base Angles Theorem, we will construct the angle bisector through the vertex angle of an isosceles triangle.
Given: Isosceles triangle
1. Isosceles triangle
2. Construct angle bisector
|Every angle has one angle bisector|
||Definition of an angle bisector|
By constructing the angle bisector,
Isosceles Triangle Theorem: The angle bisector of the vertex angle in an isosceles triangle is also the perpendicular bisector to the base.
The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true.
Base Angles Theorem Converse: If two angles in a triangle are congruent, then the opposite sides are also congruent.
So, for a triangle
Isosceles Triangle Theorem Converse: The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle.
In other words, if
Which two angles are congruent?
This is an isosceles triangle. The congruent angles, are opposite the congruent sides.
From the arrows we see that
If an isosceles triangle has base angles with measures of
Draw a picture and set up an equation to solve for the vertex angle,
If an isosceles triangle has a vertex angle with a measure of
Draw a picture and set up and equation to solve for the base angles,
Watch this video for help with the Examples above.
An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides of the isosceles triangle are called the legs. The other side is called the base. The angles between the base and the legs are called base angles. The angle made by the two legs is called the vertex angle.
1. Find the value of
2. Find the measure of
3. True or false: Base angles of an isosceles triangle can be right angles.
1. Set the angles equal to each other and solve for
2. The two sides are equal, so set them equal to each other and solve for
3. This statement is false. Because the base angles of an isosceles triangle are congruent, if one base angle is a right angle then both base angles must be right angles. It is impossible to have a triangle with two right (
Find the measures of
Determine if the following statements are true or false.
- Base angles of an isosceles triangle are congruent.
- Base angles of an isosceles triangle are complementary.
- Base angles of an isosceles triangle can be equal to the vertex angle.
- Base angles of an isosceles triangle are acute.
Complete the proofs below.
△CIS, with base angles ∠Cand ∠S IO¯¯¯¯is the angle bisector of ∠CISProve: IO¯¯¯¯is the perpendicular bisector of CS¯¯¯¯¯
△ICSwith ∠Cand ∠S IO¯¯¯¯is the perpendicular bisector of CS¯¯¯¯¯Prove: IO¯¯¯¯is the angle bisector of ∠CIS
- (-2, 1), (1, -2), (-5, -2)
- (-2, 5), (2, 4), (0, -1)
- (6, 9), (12, 3), (3, -6)
- (-10, -5), (-8, 5), (2, 3)
- (-1, 2), (7, 2), (3, 9)