Question

1. complete the table by entering the measures of the unknown angles for the following two triangles.

triangle	angle 1	angle 2	unknown angle
2	45°	20°

1. if an exterior angle of a triangle has a measure of 35°, what is the measure of the adjacent interior angle?
2. find the value of x.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. For triangle 1 with angle 1 = 30° and angle 2 = 60°, let the unknown angle be \(x\). Then \(x=180-(30 + 60)=90^{\circ}\).

Step2: Apply angle - sum property for triangle 2

For triangle 2 with angle 1 = 45° and angle 2 = 20°, let the unknown angle be \(y\). Then \(y = 180-(45+20)=115^{\circ}\).

Step3: Recall exterior - interior angle relationship

An exterior angle of a triangle and its adjacent interior angle are supplementary (sum to 180°). If the exterior angle is 35°, then the adjacent interior angle \(z=180 - 35=145^{\circ}\).

Step4: Use exterior - angle property of a triangle

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \((x + 5)=(x - 5)+(180 - 120)\). Simplify the right - hand side: \((x - 5)+60=x + 55\). Then \(x+5=x + 55\) is incorrect. Using the fact that the sum of interior angles of a triangle is 180°, we have \((x - 5)+(x + 5)+(180 - 120)=180\). Combine like terms: \(2x+60 = 180\). Subtract 60 from both sides: \(2x=180 - 60=120\). Divide both sides by 2: \(x = 60\).

Answer:

1. Triangle 1: 90°; Triangle 2: 115°
2. 145°
3. 60