Question

find the measures of the interior angles of the triangle. (see example 1.)1.2.3.4.

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).

For problem 1:

We know one angle is \(90^{\circ}\) (right - angle) and another is \(30^{\circ}\). Let the third angle be \(x\). Then \(30 + 90+x=180\).

For problem 2:

Given two angles \(65^{\circ}\) and \(40^{\circ}\), let the third angle be \(x\). Then \(65 + 40+x=180\).

For problem 3:

First, use the linear - pair relationship. The non - adjacent interior angle to \((x + 65)^{\circ}\) (linear pair) and the two given interior angles \(x^{\circ}\) and \(25^{\circ}\) satisfy the triangle angle - sum theorem. Also, the angle adjacent to \((x + 65)^{\circ}\) is \(180-(x + 65)=115 - x\). Then \(x+25+(115 - x)=180\) (which is always true, but we can also use the exterior - angle theorem. The exterior angle \((x + 65)^{\circ}\) is equal to the sum of the two non - adjacent interior angles. So \(x + 65=x + 25\) (wrong, better use \(x+(x + 65)+25=180\)).

For problem 4:

Given angles \(x^{\circ}\), \(48^{\circ}\), and \((x - 44)^{\circ}\). Then \(x+48+(x - 44)=180\).

Step2: Solve the equations

Problem 1:

$$
LATEXBLOCK0
$$

The angles are \(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\).

Problem 2:

$$
LATEXBLOCK1
$$

The angles are \(40^{\circ}\), \(65^{\circ}\), \(75^{\circ}\).

Problem 3:

$$
LATEXBLOCK2
$$

The angles are \(25^{\circ}\), \(45^{\circ}\), \(110^{\circ}\) (since \(x + 65=45+65 = 110\)).

Problem 4:

$$
LATEXBLOCK3
$$

The angles are \(48^{\circ}\), \(88^{\circ}\), \(44^{\circ}\) (since \(x-44=88 - 44\)).

Answer:

1. The measures of the interior angles are \(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\).
2. The measures of the interior angles are \(40^{\circ}\), \(65^{\circ}\), \(75^{\circ}\).
3. The measures of the interior angles are \(25^{\circ}\), \(45^{\circ}\), \(110^{\circ}\).
4. The measures of the interior angles are \(48^{\circ}\), \(88^{\circ}\), \(44^{\circ}\).