Question

use what you know about the angles of a triangle to find the value of x and the angles in each triangle below.
use the triangle angle sum theorem, explained in the math notes box in lesson 2.2.1.
hint (a):
( x + 79 ^ { circ } + 90 ^ { circ } = 180 ^ { circ } )
hint (b):
( x + x + 90 ^ { circ } = 180 ^ { circ } )
answer (c):
answer (d):

Explanation:

Step1: Recall the Triangle Angle Sum Theorem

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

Step2: Solve for \(x\) in part (b)

Since it is a right - angled triangle (one angle is \(90^{\circ}\)) and the two non - right angles are equal (\(x\) each), we have the equation \(x + x+90^{\circ}=180^{\circ}\).
Combine like terms: \(2x+90^{\circ}=180^{\circ}\).
Subtract \(90^{\circ}\) from both sides: \(2x=180^{\circ}-90^{\circ}=90^{\circ}\).
Divide both sides by 2: \(x = 45^{\circ}\).

Step3: Solve for \(x\) in part (c)

It is a right - angled triangle (one angle is \(90^{\circ}\)) and another angle is \(27^{\circ}\). Using the Triangle Angle Sum Theorem \(x + 27^{\circ}+90^{\circ}=180^{\circ}\).
Combine like terms: \(x+117^{\circ}=180^{\circ}\).
Subtract \(117^{\circ}\) from both sides: \(x=180^{\circ}-117^{\circ}=63^{\circ}\).

Step4: Solve for \(x\) in part (d)

It is a right - angled triangle (one angle is \(90^{\circ}\)) and another angle is \(22^{\circ}\). Using the Triangle Angle Sum Theorem \(x + 22^{\circ}+90^{\circ}=180^{\circ}\).
Combine like terms: \(x + 112^{\circ}=180^{\circ}\).
Subtract \(112^{\circ}\) from both sides: \(x=180^{\circ}-112^{\circ}=68^{\circ}\).

Answer:

b. \(x = 45^{\circ}\)
c. \(x = 63^{\circ}\)
d. \(x = 68^{\circ}\)