This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for the unknown angles in each diagram: a) Step 1: Identify the type of triangle. The two tick marks on the sides indicate that it is an isosceles triangle. The angles opposite the equal sides are equal. b = 70^ Step 2: Use the sum of angles in a triangle. The sum of interior angles in a triangle is 180^. a + b + 70^ = 180^ a + 70^ + 70^ = 180^ a + 140^ = 180^ a = 180^ - 140^ a = 40^ The unknown angles are a = 40^, b = 70^. b) Step 1: Find the interior angle q. The sum of interior angles in a triangle is 180^. q + 60^ + 60^ = 180^ q + 120^ = 180^ q = 180^ - 120^ q = 60^ Step 2: Find the exterior angle p. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. p = 60^ + 60^ p = 120^ Alternatively, p and q form a linear pair, so p + q = 180^. p + 60^ = 180^ p = 180^ - 60^ p = 120^ The unknown angles are p = 120^, q = 60^. c) Step 1: Identify the type of triangle. The tick marks on the sides and the angle markings indicate that all three sides are equal and all three angles are equal. This is an equilateral triangle. Step 2: Determine the angles. In an equilateral triangle, each interior angle is 60^. x = 60^ z = 60^ The unknown angles are x = 60^, z = 60^. d) Step 1: Find the interior angle adjacent to the 130^ exterior angle. Let this interior angle be y. Angles on a straight line sum to 180^. y + 130^ = 180^ y = 180^ - 130^ y = 50^ Step 2: Use the sum of angles in a triangle. The sum of interior angles in a triangle is 180^. b + 60^ + y = 180^ b + 60^ + 50^ = 180^ b + 110^ = 180^ b = 180^ - 110^ b = 70^ Alternatively, the exterior angle is equal to the sum of the two opposite interior angles. 130^ = b + 60^ b = 130^ - 60^ b = 70^ The unknown angle is b = 70^. e) Step 1: Identify the type of triangle. The square symbol indicates a right angle, which is 90^. Step 2: Use the sum of angles in a triangle. The sum of interior angles in a triangle is 180^. a + 90^ + 50^ = 180^ a + 140^ = 180^ a = 180^ - 140^ a = 40^ The unknown angle is a = 40^. f) Step 1: Use the sum of angles in a triangle. The sum of interior angles in a triangle is 180^. x + 60^ + 40^ = 180^ x + 100^ = 180^ x = 180^ - 100^ x = 80^ The unknown angle is x = 80^. g) Step 1: Find angle z. Angle z and the 140^ exterior angle form a linear pair. z + 140^ = 180^ z = 180^ - 140^ z = 40^ Step 2: Identify the type of triangle. The tick marks on the sides indicate that it is an isosceles triangle. The angles opposite the equal sides are equal, so x = y. Step 3: Use the sum of angles in a triangle. x + y + z = 180^ x + x + 40^ = 180^ 2x + 40^ = 180^ 2x = 180^ - 40^ 2x = 140^ x = (140^)/(2) x = 70^ Since x = y, then y = 70^. The unknown angles are x = 70^, y = 70^, z = 40^. h) Step 1: Find angle z. Angle z and the 80^ exterior angle form a linear pair. z + 80^ = 180^ z = 180^ - 80^ z = 100^ Step 2: Identify the type of triangle. The tick marks on the sides indicate that it is an isosceles triangle. The angles opposite the equal sides are equal, so x = y. Step 3: Use the sum of angles in a triangle. x + y + z = 180^ x + x + 100^ = 180^ 2x + 100^ = 180^ 2x = 180^ - 100^ 2x = 80^ x = (80^)/(2) x = 40^ Since x = y, then y = 40^. The unknown angles are x = 40^, y = 40^, z = 100^. i) Step 1: Find angle w in triangle ADC. The sum of angles in triangle ADC is 180^. w + 45^ + 30^ = 180^ w + 75^ = 180^ w = 180^ - 75^ w = 105^ Step 2: Find angle y in triangle ABC. The sum of angles in triangle ABC is 180^. y + 60^ + 50^ = 180^ y + 110^ = 180^ y = 180^ - 110^ y = 70^ Step 3: Find angle x. Angle x is the sum of the two angles at vertex A. x = 45^ + 60^ x = 105^ Step 4: Find angle z. Angle z is the sum of the two angles at vertex C. z = 30^ + 50^ z = 80^ The unknown angles are x = 105^, y = 70^, z = 80^, w = 105^. Send me the next one 📸