The values of the marked angles are:
11) ∠1 = 81⁰ ∠4 = 74⁰
∠2 = 24⁰ ∠5 = 24⁰
∠3 = 105⁰
12) ∠1 = 76⁰ ∠3 = 76⁰
∠2 = 104⁰ ∠4 = 80⁰
What are similar triangles?
In similar triangles, corresponding angles are equal, meaning that each angle in one triangle has a corresponding angle in the other triangle with the same measure.
11) The two triangles in the figures are similar.
∠4 = 74⁰(corresponding ∠s of similar ∆)
∠3 + 75 = 180⁰( linear ∠s)
∠3 = 180 - 75
∠3 = 105⁰
∠2 + 156 = 180(linear ∠s)
∠2 = 180 - 156
= 24⁰
∠5 = 24⁰(corresponding ∠s of similar ∆)
75 + 24 + ∠1 = 180(sum of int. ∠s of a ∆)
99 + ∠1 = 180
∠1 = 180 - 99 = 81⁰
12) 16x + 20 + 21x - 25 = 180(adjacent angles of quadrilateral
37x = 180 + 5
37x = 185
x = 185/37 = 5
21x -25 = 21(5) -25
= 105 - 25 = 80⁰
16x + 20 = 16(5) + 20
= 80 + 20 = 100⁰
∠4 = 80⁰( corresponding ∠s of similar ∆)
∠1 + 80 + 24 = 180(∠s in a ∆)
∠1 + 104 = 180
∠1 = 180 - 104 = 76⁰
∠1 + ∠2 = 180(linear ∠s)
∠2 = 180 - 76 = 104⁰
∠1 = ∠3 = 76⁰