The exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles. Let the unknown interior angle be x.

Here are the solutions to the problems: 1. An exterior angle of a triangle is 125^. If one of the non-adjacent interior angles is 48^, find the other non-adjacent interior angle. Step 1: The exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles. Let the unknown interior angle be x. 125^ = 48^ + x Step 2: Subtract 48^ from both sides to find x. x = 125^ - 48^ x = 77^ The other non-adjacent interior angle is 77^ 2. A regular polygon has an interior angle of 144^. a) Explain why this polygon must have 10 sides. Step 1: Find the measure of one exterior angle. The interior angle and exterior angle at any vertex of a polygon are supplementary. Exterior angle = 180^ - Interior angle Exterior angle = 180^ - 144^ Exterior angle = 36^ Step 2: For any regular polygon, the sum of its exterior angles is 360^. If n is the number of sides, then each exterior angle is (360^)/(n). (360^)/(n) = 36^ Step 3: Solve for n. n = (360^)/(36^) n = 10 Therefore, the polygon must have 10 sides. b) Justify your reasoning using the formula for the sum of interior angles: Sum = (n-2) × 180^, where n is the number of sides. Step 1: For a regular polygon, each interior angle is equal to the sum of interior angles divided by the number of sides, n. Each interior angle = ((n-2) × 180^)/(n) Step 2: Set the given interior angle equal to the formula and solve for n. 144^ = ((n-2) × 180^)/(n) Step 3: Multiply both sides by n. 144n = (n-2) × 180 144n = 180n - 360 Step 4: Rearrange the equation to solve for n. 360 = 180n - 144n 360 = 36n n = (360)/(36) n = 10 The polygon has 10 sides 3. Two angles are supplementary. One angle measures 67^. What is the measure of the other angle? Step 1: Supplementary angles add up to 180^. Let the unknown angle be x. 67^ + x = 180^ Step 2: Subtract 67^ from both sides to find x. x = 180^ - 67^ x = 113^ The measure of the other angle is 113^ 4. A right-angled triangle has sides of 5 cm and 12 cm. Using the Pythagorean theorem, calculate the length of the hypotenuse. Step 1: The Pythagorean theorem states that for a right-angled triangle, a^2 + b^2 = c^2, where a and b are the lengths of the two shorter sides (legs), and c is the length of the hypotenuse. Let a = 5 cm and b = 12 cm. 5^2 + 12^2 = c^2 Step 2: Calculate the squares of the sides. 25 + 144 = c^2 Step 3: Add the values and find the square root to get c. 169 = c^2 c = sqrt(169) c = 13 cm The length of the hypotenuse is 13 cm 5. A ladder is leaning against a wall. The ladder is 10 metres long, and the base of the ladder is 6 metres away from the wall. a) Calculate the height the ladder reaches on the wall. Step 1: This scenario forms a right-angled triangle. The ladder is the hypotenuse (c), the distance from the wall is one leg (b), and the height the ladder reaches on the wall is the other leg (a). Using the Pythagorean theorem: a^2 + b^2 = c^2. Given: c = 10 m, b = 6 m. We need to find a. a^2 + 6^2 = 10^2 Step 2: Calculate the squares. a^2 + 36 = 100 Step 3: Subtract 36 from both sides. a^2 = 100 - 36 a^2 = 64 Step 4: Take the square root to find a. a = sqrt(64) a = 8 m The height the ladder reaches on the wall is 8 m What's next? Send 'em! 📸