# In the trapezoid ABCD AB = CD, AC = AD and ∠ABC = 95 °. Find the CAD angle.

May 31, 2020 | Education

| Since AB = CD, then the trapezoid ABCD is isosceles.

Then, by the property of an isosceles trapezoid, ∠ABC = ∠BCD = 95 ° and ∠CDA = ∠DAB.

Recalling that the sum of the angles of a convex n-gon is calculated by the formula (n-2) 180 °, we find that the sum of the angles of the trapezoid is (4-2) 180 ° = 360 °.

Then ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360 °

95 ° + 95 ° + ∠CDA + ∠DAB = 360 °

∠CDA + ∠DAB = 170 °

∠CDA = ∠DAB = 170 ° / 2 = 85 °

Consider the triangle ACD.

Since AC = AD, this triangle is isosceles.

Therefore, by the property of an isosceles triangle, ∠CDA = ∠DCA = 85 °

∠BCA = ∠BCD-∠DCA = 95 ° -85 ° = 10 °

∠CAD = ∠DCA = 10 ° (because they are cross-lying for parallel lines AD and BC).

Answer: 10

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