Question

1. determine the measure of the angles.

a) m∠1 =
m∠2 =
b) m∠1 =
m∠2 =
c) m∠1 =
m∠2 =

1. determine the missing information.

a) x =
m∠a =
b) x =
m∠b =
c) x =
m∠a =

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°.

Step2: Solve for the first triangle in part 1 (top - left)

We have the equation \(2x+(3x - 3)+68 = 180\). Combine like - terms: \(5x+65 = 180\). Subtract 65 from both sides: \(5x=180 - 65=115\). Divide both sides by 5: \(x = 23\). Then \(m\angle A=3x - 3=3\times23 - 3=69 - 3 = 66^{\circ}\).

Step3: Solve for the second triangle in part 1 (middle - left)

The equation for the sum of angles is \(2x+3x + 45=180\). Combine like - terms: \(5x+45 = 180\). Subtract 45 from both sides: \(5x=180 - 45 = 135\). Divide by 5: \(x = 27\). Then \(m\angle B=3x=3\times27 = 81^{\circ}\).

Step4: Solve for the third triangle in part 1 (bottom - left)

Since the triangle is isosceles (two equal sides), the base angles are equal. The equation is \(2x+(3x + 5)+2x=180\). Combine like - terms: \(7x+5 = 180\). Subtract 5 from both sides: \(7x=175\). Divide by 7: \(x = 25\). Then \(m\angle A=3x + 5=3\times25+5=75 + 5=80^{\circ}\).

Step5: Solve for the first triangle in part 2 (top - right)

In the larger triangle, one angle is \(70^{\circ}\) and another is \(47^{\circ}\), so the third angle (not labeled) is \(180-(70 + 47)=63^{\circ}\). Since \(\angle1\) and the non - labeled angle are vertical angles, \(m\angle1 = 63^{\circ}\). In the smaller triangle with \(\angle2\), we know one angle is \(47^{\circ}\) and \(\angle1 = 63^{\circ}\), so \(m\angle2=180-(47 + 63)=70^{\circ}\).

Step6: Solve for the second triangle in part 2 (middle - right)

The large triangle has an angle of \(64^{\circ}\). Since the two smaller triangles are isosceles (marked by equal sides), in the upper small isosceles triangle, \(m\angle1=\frac{180 - 64}{2}=58^{\circ}\). In the lower small isosceles triangle, \(m\angle2 = 58^{\circ}\) (corresponding angles). And \(m\angle3=64^{\circ}\) (vertical angles with the \(64^{\circ}\) angle in the large triangle).

Step7: Solve for the third triangle in part 2 (bottom - right)

The large triangle has an angle of \(60^{\circ}\). Since the two smaller triangles are isosceles (marked by equal sides), in the upper small isosceles triangle, \(m\angle1=\frac{180 - 60}{2}=60^{\circ}\). In the lower small isosceles triangle, \(m\angle2 = 60^{\circ}\) (corresponding angles). And \(m\angle3=60^{\circ}\) (vertical angles with the \(60^{\circ}\) angle in the large triangle).

Answer:

1.
a) \(x = 23\), \(m\angle A=66^{\circ}\)
b) \(x = 27\), \(m\angle B=81^{\circ}\)
c) \(x = 25\), \(m\angle A=80^{\circ}\)
2.
a) \(m\angle1 = 63^{\circ}\), \(m\angle2 = 70^{\circ}\)
b) \(m\angle1 = 58^{\circ}\), \(m\angle2 = 58^{\circ}\), \(m\angle3=64^{\circ}\)
c) \(m\angle1 = 60^{\circ}\), \(m\angle2 = 60^{\circ}\), \(m\angle3=60^{\circ}\)