The isosceles triangle: the converse theorem

Geometry, grade 9 Lesson 1 of 21Learning

The isosceles triangle: the converse theorem

The base angles of an isosceles triangle

In eighth grade you learned: in an isosceles triangle the base angles are equal – the angles facing the legs.

Today the other way round: two equal angles are known, and two equal sides follow.

The converse theorem

The equal angles are the base angles, and the side between them is the base.

Theorem: a triangle with two equal angles is isosceles: the sides facing the equal angles are equal.

In triangle $\displaystyle ABC$ , if $\displaystyle {\angle B}={\angle C}$ , then $\displaystyle {AB}={AC}$ .
$\displaystyle AB$ faces $\displaystyle \angle C$ , and $\displaystyle AC$ faces $\displaystyle \angle B$ .

Proving the converse theorem

Given: $\displaystyle {\angle B}={\angle C}$ .
Prove: $\displaystyle {AB}={AC}$ .

The proof below draws $\displaystyle AD$ , the bisector of angle $\displaystyle A$ , and shows that triangles $\displaystyle ABD$ and $\displaystyle ACD$ are congruent:

	Claim	Reason
1	$\displaystyle {\angle B}={\angle C}$	Given
2	$\displaystyle {\angle BAD}={\angle CAD}$	$\displaystyle AD$ bisects angle $\displaystyle A$ (by construction)
3	$\displaystyle {\angle ADB}={\angle ADC}$	(1), (2); the angles of a triangle add up to $\displaystyle 180^{\circ}$
4	$\displaystyle {AD}={AD}$	A common side
5	$\displaystyle \triangle ABD\cong\triangle ACD$	(2), (3), (4); the congruence theorem ASA
6	$\displaystyle {AB}={AC}$	(5); in congruent triangles, equal angles face equal sides

A triangle with three equal angles

When all three angles are equal, the theorem holds for each pair of them, so all the sides are equal: a triangle with three equal angles is equilateral. Each of its angles is $\displaystyle 60^{\circ}$ , since $\displaystyle {180^{\circ}:3}={60^{\circ}}$ .

Finding a side from the angles

First find the equal angles, then the sides facing them.

In triangle $\displaystyle ABC$ , $\displaystyle {\angle A}={80^{\circ}}$ , $\displaystyle {\angle B}={50^{\circ}}$ , and $\displaystyle {AC}={7}$ :

$\displaystyle {\angle C}={180^{\circ}-80^{\circ}-50^{\circ}}={50^{\circ}}$ , so $\displaystyle {\angle B}={\angle C}$ ;
therefore $\displaystyle {AB}={AC}={7}$ : $\displaystyle AB$ faces $\displaystyle \angle C$ , and $\displaystyle AC$ faces $\displaystyle \angle B$ .

Exercise 1

Which sides of the triangle are equal? Write the sides and give the reason.

In triangle $\displaystyle KLM$ , $\displaystyle {\angle K}={40^{\circ}}$ and $\displaystyle {\angle L}={70^{\circ}}$ .

In triangle $\displaystyle PQR$ , angle $\displaystyle P$ is right and $\displaystyle {\angle Q}={45^{\circ}}$ .

Exercise 2

In triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
$\displaystyle BD$ bisects angle $\displaystyle B$ , and $\displaystyle CE$ bisects angle $\displaystyle C$ .
They meet at the point $\displaystyle O$ .
Prove: $\displaystyle {OB}={OC}$ .

Exercise 3

In triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
The segment $\displaystyle DE$ is parallel to $\displaystyle BC$ : $\displaystyle D$ is on $\displaystyle AB$ , and $\displaystyle E$ on $\displaystyle AC$ .
Prove: $\displaystyle {AD}={AE}$ .

Formulas: geometry, grade 9

Open as a page

Angles and sides in a triangle

The sum of the angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
An exterior angle $\displaystyle \delta$ equals the sum of the two interior angles not next to it
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
The larger side faces the larger angle, and the other way round
$\displaystyle a>b\ \Leftrightarrow\ \alpha>\beta$

The right triangle

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^{2}+b^{2}$ $\displaystyle =$ $\displaystyle c^{2}$
The leg $\displaystyle a$ facing a $\displaystyle 30^{\circ}$ angle is half the hypotenuse $\displaystyle c$
$\displaystyle a$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$
The median $\displaystyle m_c$ to the hypotenuse is half the hypotenuse $\displaystyle c$
$\displaystyle m_c$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$

Angles between parallel lines

Corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Co-interior angles
$\displaystyle \alpha+\beta=180^{\circ}$

Angles in a quadrilateral

The sum of the angles of a quadrilateral
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$
In a trapezoid: the two angles beside a leg
$\displaystyle \alpha+\beta=180^{\circ}$

Areas

The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a rectangle
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^{2}$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a rhombus or a kite – $\displaystyle d_1$ , $\displaystyle d_2$ the diagonals
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{d_1\cdot d_2}{2}$
The ratio of the areas of similar triangles – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$