Question

9. If one angle of a triangle is equal to the sum of the other two, show that the triangle is
right-angled.​

Answer 1

Things to know

• Triangle angle sum (Sol.1)

→ All angles in a triangle sum to 180°.

• Interior and exterior angle (Sol.2)

→ An interior angle and an exterior angle are on a line.

Solution

Sol.1

First, let's assume two angles are a° and b°.

The measure of the remaining angle will be $(180-a-b)\textdegree$. It's because all angles in a triangle sum to 180°.

According to the question,

$a+b=180-a-b$

$\Leftrightarrow 2a+2b=180$

$\Leftrightarrow a+b=90\textdegree$

The sum of two angles is $90\textdegree$ and one remaining angle is $90\textdegree$. The triangle is right-angled.

Sol.2

(Attachment 1)

Again, let's assume two angles are a° and b°. Then let's extend one side of the triangle.

The sum of the two angles is the exterior angle and the remaining angle is the interior angle.

Since they are on a line, for them to be equal each needs to be $90\textdegree$.

The triangle is right-angled.

Why?

Why do all angles in a triangle sum to 180°?

(Attachment 2)

All angles in a triangle sum to 180°. To show is not difficult.

First, let's draw a parallel line against a side, passing through one vertice.

Through using parallel lines, we can show the angle sum is 180°.

Answer 2

Answer:

Things to know

Triangle angle sum (Sol.1)

→ All angles in a triangle sum to 180°.

Interior and exterior angle (Sol.2)

→ An interior angle and an exterior angle are on a line.

Solution

Sol.1

First, let's assume two angles are a° and b°.

The measure of the remaining angle will be (180-a-b)\textdegree(180−a−b)\textdegree . It's because all angles in a triangle sum to 180°.

According to the question,

a+b=180-a-ba+b=180−a−b

\Leftrightarrow 2a+2b=180⇔2a+2b=180

\Leftrightarrow a+b=90\textdegree⇔a+b=90\textdegree

The sum of two angles is 90\textdegree90\textdegree and one remaining angle is 90\textdegree90\textdegree . The triangle is right-angled.

Sol.2

(Attachment 1)

Again, let's assume two angles are a° and b°. Then let's extend one side of the triangle.

The sum of the two angles is the exterior angle and the remaining angle is the interior angle.

Since they are on a line, for them to be equal each needs to be 90\textdegree90\textdegree .

The triangle is right-angled.

Why?

Why do all angles in a triangle sum to 180°?

(Attachment 2)

All angles in a triangle sum to 180°. To show is not difficult.

First, let's draw a parallel line against a side, passing through one vertice.

Through using parallel lines, we can show the angle sum is 180°.