Question

isosceles and equilateral triangles
find the missing angle measures.
1
find the value of each variable.
4
7
8
9

Explanation:

Step1: Use the property of isosceles triangle (equal sides have equal angles)

For problem 1: In \(\triangle MNP\), since \(MP = NP\), then \(m\angle M=m\angle N\). Given \(m\angle M = 64^{\circ}\), so \(m\angle N=64^{\circ}\).

Step2: Use the angle - sum property of a triangle (\(m\angle A + m\angle B+m\angle C=180^{\circ}\))

For problem 2: In \(\triangle DEF\), \(m\angle D=m\angle F = 33^{\circ}\). Then \(m\angle E=180-(33 + 33)=114^{\circ}\).

Step3: Use the property of isosceles triangle and angle - sum property

For problem 3: In \(\triangle ABC\), \(m\angle A=m\angle C\). Let \(m\angle A = m\angle C=x\). Then \(x + x+40=180\), \(2x=140\), \(x = 70^{\circ}\), so \(m\angle A=70^{\circ}\).

Step4: Use the property of right - isosceles triangle (in a right - isosceles triangle, the two non - right angles are equal)

For problem 4: In the right - triangle (right - angle \(=90^{\circ}\)), \(x + y+90 = 180\) and \(x=y\) (because the two non - hypotenuse sides are equal). Then \(2x=90\), \(x = 45\), \(y = 45\).

Step5: Use the property of equilateral triangle (all angles are equal, \(60^{\circ}\))

For problem 5: Since it is an equilateral triangle (all sides are equal), \(10x=60\), \(x = 6\); \(12y=60\), \(y = 5\).

Step6: Use the property of isosceles triangle and angle - sum property

For problem 6: In the isosceles triangle (two sides are equal), \(x+71 + y=180\) and \(x=y\). Then \(2x=109\), \(x = 71\), \(y = 38\).

Step7: Use the property of isosceles triangle and angle - sum property

For problem 7: The large isosceles triangle has \(y = 70^{\circ}\) (equal sides). The smaller isosceles triangle: \(x=180-(70 + 70)=40^{\circ}\).

Step8: Use the property of isosceles triangle (angle bisector in an isosceles triangle)

For problem 8: The angle \(52^{\circ}\) is bisected (because of the equal sides). So \(x = 26\), \(y=26\).

Step9: Use the property of equilateral triangle (all angles \(60^{\circ}\))

For problem 9: \(20x=60\), \(x = 3\); \(5y=60\), \(y = 12\); \(15z=60\), \(z = 4\).

Answer:

1. \(m\angle N = 64^{\circ}\)
2. \(m\angle E=114^{\circ}\)
3. \(m\angle A = 70^{\circ}\)
4. \(x = 45\), \(y = 45\)
5. \(x = 6\), \(y = 5\)
6. \(x = 71\), \(y = 38\)
7. \(x = 40\), \(y = 70\)
8. \(x = 26\), \(y = 26\)
9. \(x = 3\), \(y = 12\), \(z = 4\)