ngles a 61 . One angle in a triangle is a right angle, and one angle is equal to \( 30^{\circ} \). What is the measure of the third angle? 62. A triangle has a \( 45^{\circ} \) angle and a right angle. Find the measure of the third angle. 63. Two angles of a triangle measure \( 42^{\circ} \) and \( 103^{\circ} \). Find the measure of the third angle. 64. Two angles of a triangle measure \( 62^{\circ} \) and \( 45^{\circ} \). Find the measure of the third angle. 65. A triangle has a \( 13^{\circ} \) angle and a \( 65^{\circ} \) angle. What is the measure of the third angle? 66. A triangle has a \( 105^{\circ} \) angle and a \( 32^{\circ} \) angle. What is the measure of the third angle? b7. Cut out a triangle and then tear off two of the angles, as shown below. Position the pieces you tore off so that angle a a adjacent to angle \( b \) and angle cis adjacent to angle \( b \). Describe what you observe. What does this demonstrete?

To find the measure of the third angle in each triangle, we can use the fact that the sum of the angles in a triangle is always \( 180^{\circ} \). ### 61. One angle is a right angle and one angle is \( 30^{\circ} \). - Right angle = \( 90^{\circ} \) - Second angle = \( 30^{\circ} \) - Third angle = \( 180^{\circ} - (90^{\circ} + 30^{\circ}) = 180^{\circ} - 120^{\circ} = 60^{\circ} \) **Third angle: \( 60^{\circ} \)** ### 62. One angle is \( 45^{\circ} \) and one angle is a right angle. - Right angle = \( 90^{\circ} \) - Second angle = \( 45^{\circ} \) - Third angle = \( 180^{\circ} - (90^{\circ} + 45^{\circ}) = 180^{\circ} - 135^{\circ} = 45^{\circ} \) **Third angle: \( 45^{\circ} \)** ### 63. Two angles measure \( 42^{\circ} \) and \( 103^{\circ} \). - First angle = \( 42^{\circ} \) - Second angle = \( 103^{\circ} \) - Third angle = \( 180^{\circ} - (42^{\circ} + 103^{\circ}) = 180^{\circ} - 145^{\circ} = 35^{\circ} \) **Third angle: \( 35^{\circ} \)** ### 64. Two angles measure \( 62^{\circ} \) and \( 45^{\circ} \). - First angle = \( 62^{\circ} \) - Second angle = \( 45^{\circ} \) - Third angle = \( 180^{\circ} - (62^{\circ} + 45^{\circ}) = 180^{\circ} - 107^{\circ} = 73^{\circ} \) **Third angle: \( 73^{\circ} \)** ### 65. One angle is \( 13^{\circ} \) and one angle is \( 65^{\circ} \). - First angle = \( 13^{\circ} \) - Second angle = \( 65^{\circ} \) - Third angle = \( 180^{\circ} - (13^{\circ} + 65^{\circ}) = 180^{\circ} - 78^{\circ} = 102^{\circ} \) **Third angle: \( 102^{\circ} \)** ### 66. One angle is \( 105^{\circ} \) and one angle is \( 32^{\circ} \). - First angle = \( 105^{\circ} \) - Second angle = \( 32^{\circ} \) - Third angle = \( 180^{\circ} - (105^{\circ} + 32^{\circ}) = 180^{\circ} - 137^{\circ} = 43^{\circ} \) **Third angle: \( 43^{\circ} \)** ### 67. Observations from tearing off two angles of a triangle: When you cut out a triangle and tear off two angles, and then position them adjacent to each other, you will observe that the two angles together will form a straight line (i.e., they will sum to \( 180^{\circ} \)). This demonstrates that the sum of the angles in a triangle is equal to \( 180^{\circ} \). This is a fundamental property of triangles in Euclidean geometry.