Vertical Angles and Linear Pairs Previously, you learned that two angles are adjacent if they share a common vertex and side but have no common interior points. In this lesson, you will study other relationships between pairs of angles. Two angles are vertical angles if their sides form two pairs of opposite rays. 1 4 3 2 1 and 3 are vertical angles. 2 and 4 are vertical angles.

Vertical Angles and Linear Pairs Previously, you learned that two angles are adjacent if they share a common vertex and side but have no common interior points. In this lesson, you will study other relationships between pairs of angles. Two angles are vertical angles if their sides form two pairs of opposite rays. Two adjacent angles are a linear pair if their noncommon sides are opposite rays. 1 4 2 5 6 3 1 and 3 are vertical angles. 5 and 6 are a linear pair. 2 and 4 are vertical angles.

Identifying Vertical Angles and Linear Pairs Answer the questions using the diagram. Are 2 and 3 a linear pair? 1 2 4 3 SOLUTION No. The angles are adjacent but their noncommon sides are not opposite rays.

Identifying Vertical Angles and Linear Pairs Answer the questions using the diagram. Are 2 and 3 a linear pair? 1 2 4 3 Are 3 and 4 a linear pair? SOLUTION No. The angles are adjacent but their noncommon sides are not opposite rays. Yes. The angles are adjacent and their noncommon sides are opposite rays.

Identifying Vertical Angles and Linear Pairs Answer the questions using the diagram. Are 2 and 3 a linear pair? 1 2 4 3 Are 3 and 4 a linear pair? Are 1 and 3 vertical angles? SOLUTION No. The angles are adjacent but their noncommon sides are not opposite rays. Yes. The angles are adjacent and their noncommon sides are opposite rays. No. The sides of the angles do not form two pairs of opposite rays.

Identifying Vertical Angles and Linear Pairs Answer the questions using the diagram. Are 2 and 3 a linear pair? 1 2 4 3 Are 3 and 4 a linear pair? Are 1 and 3 vertical angles? Are 2 and 4 vertical angles? SOLUTION No. The angles are adjacent but their noncommon sides are not opposite rays. Yes. The angles are adjacent and their noncommon sides are opposite rays. No. The sides of the angles do not form two pairs of opposite rays. No. The sides of the angles do not form two pairs of opposite rays.

Finding Angle Measures In the stair railing shown, 6 has a measure of 130˚. Find the measures of the other three angles. SOLUTION 5 6 7 8 6 and 7 are a linear pair. So, the sum of their measures is 180˚. m6 + m7 = 180˚ 130˚ + m7 = 180˚ m7 = 50˚

Finding Angle Measures In the stair railing shown, 6 has a measure of 130˚. Find the measures of the other three angles. SOLUTION 5 6 7 8 6 and 7 are a linear pair. So, the sum of their measures is 180˚. m6 + m7 = 180˚ 130˚ + m7 = 180˚ m7 = 50˚ 6 and 5 are also a linear pair. So it follows that m5 = 50˚.

Finding Angle Measures In the stair railing shown, 6 has a measure of 130˚. Find the measures of the other three angles. SOLUTION 5 6 7 8 6 and 8 are vertical angles. So, they are congruent and have the same measure. m 8 = m 6 = 130˚

• B Finding Angle Measures Solve for x and y. Then find the angle measure. ( x + 15)˚ ( 3x + 5)˚ ( y + 20)˚ ( 4y – 15)˚ D • C • B A • E Use the fact that the sum of the measures of angles that form a linear pair is 180˚. SOLUTION Use substitution to find the angle measures (x = 40, y = 35). m AED = ( 3 x + 15)˚ = (3 • 40 + 5)˚ = 125˚ m AED + m DEB = 180° m AEC + mCEB = 180° m DEB = ( x + 15)˚ = (40 + 15)˚ = 55˚ ( 3x + 5)˚ + ( x + 15)˚ = 180° ( y + 20)˚ + ( 4y – 15)˚ = 180° m AEC = ( y + 20)˚ = (35 + 20)˚ = 55˚ 4x + 20 = 180 5y + 5 = 180 m CEB = ( 4 y – 15)˚ = (4 • 35 – 15)˚ 4x = 160 = 125˚ 5y = 175 So, the angle measures are 125˚, 55˚, 55˚, and 125˚. Because the vertical angles are congruent, the result is reasonable. x = 40 y = 35

Complementary and Supplementary Angles Two angles are complementary angles if the sum of their measurements is 90˚. Each angle is the complement of the other. Complementary angles can be adjacent or nonadjacent. 1 2 3 4 complementary adjacent complementary nonadjacent

Complementary and Supplementary Angles Two angles are supplementary angles if the sum of their measurements is 180˚. Each angle is the supplement of the other. Supplementary angles can be adjacent or nonadjacent. 7 8 5 6 supplementary adjacent supplementary nonadjacent

Identifying Angles State whether the two angles are complementary, supplementary, or neither. SOLUTION The angle showing 4:00 has a measure of 120˚ and the angle showing 10:00 has a measure of 60˚. Because the sum of these two measures is 180˚, the angles are supplementary.

Finding Measures of Complements and Supplements Find the angle measure. Given that  A is a complement of C and m A = 47˚, find mC. SOLUTION mC = 90˚ – m A = 90˚ – 47˚ = 43˚

Finding Measures of Complements and Supplements Find the angle measure. Given that  A is a complement of C and m A = 47˚, find mC. Given that P is a supplement of R and mR = 36˚, find mP. SOLUTION mC = 90˚ – m A mP = 180˚ – mR = 90˚ – 47˚ = 180 ˚ – 36˚ = 43˚ = 144˚

Finding the Measure of a Complement W and  Z are complementary. The measure of  Z is 5 times the measure of W. Find m W SOLUTION Because the angles are complementary, m W + m  Z = 90˚. But m  Z = 5( m W ), so m W + 5( m W) = 90˚. Because 6(m W) = 90˚, you know that m W = 15˚.

1.7 Introduction to Perimeter, Circumference, & Area Geometry

Standards/Objectives: Standard 3: Students will understand geometric concepts and applications. Objectives: Find the perimeter and area of common plane figures. Use a general problem-solving plan.

Rectangle Perimeter P=2l +2w Area A=lw or bh W or h (width) L or b (length)

Square Perimeter P=4s Area A=s2 S (side)

Triangle Perimeter P=a+b+c Area A= ½ bh a c h (height) b (base)

Circle Diameter d=2r Circumference C=2πr Area A=πr2 radius ** Always use the π button on the calculator; even when the directions say to use 3.14.

Example: Find the perim. & area of the figure. P=2l +2w P=2(5in)+2(3in) P=10in+6in P=16in A=lw A=(5in)(3in) A=15in2 5 in 3 in

Ex: Find the perim. & area of the figure. P=4s P=4(20m) P=80m A=s2 A=(20m)2 A=400m2 20 m

Ex: Find the perim. & area of the figure. P=a+b+c P=5ft+7ft+6ft P=18 ft A= ½ bh A= ½ (7ft)(4ft) A= ½ (28ft2) A= 14ft2 5 ft 4 ft 7 ft 6 ft

Ex: Find the circumference & area of the circle. C=2πr C=2π(5in) C=10π in C≈31.4 in A=πr2 A=π(5in)2 A=25π in2 A≈78.5 in2 10in

Reminders: Chapter one Test coming up!!