Here are two different patterns made out of the same five identical isosceles triangles. Without using a protractor, determine the measures of ∠x and ∠y. Explain or show your reasoning.

The missing figure is attached

∠x = 72° and ∠y = 54°

Step-by-step explanation:

From the first patterns

∵ x is the vertex angle of each isosceles triangle

∵ All the vertices angles of the 5 Δs meet at one point

∵ The sum of the measures of angles around a point is 360°

∴ x + x + x + x + x = 360°

∴ 5 x = 360°

- Divide both sides by 5

∴ x = 72

∴ ∠x = 72°

The sum of the measures of the interior angle of a triangle is 180°

∵ The measure of the vertex angle of each Δ is x

∵ The measures of the base angles of each Δ is y

∴ x + y + y = 180°

- Add like terms in the left hand side

∴ x + 2 y = 180°

∵ x = 72°

∴ 72 + 2 y = 180

- Subtract 72 from both sides

∴ 2 y = 108

- Divide both sides by 2

∴ y = 54

∴ ∠y = 54°

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