Transcript Lesson 2-5
Warm-up 2-5 Warmup Find the value of each variable. 1. 2. 3. yo 55o 130o xo zo Fill in the blank. 4. Perpendicular lines are two lines that intersect to form __?___. 5. An angle is formed by two rays with the same endpoint. The endpoint is called the ___?___ of the angle. Lesson 2-5: Angle Relationships Term Vertical angles Adjacent angles Own Words Definition Two angles whose sides are opposite rays Two angles with one common side, a common endpoint, and share no interior points. Complementary angles Two angles whose measures sum to 90 degrees Supplementary angles Two angles whose measures sum to 180 degrees Drawing • Vertical angles 1 2 is vertical to is vertical to 3 1 4 4 2 3 Think of vertical angles as “opposite” Measures of vertical angles are equal m2 m4 m1 m3 • Adjacent angles 1 is adjacent to Adjacent means “next to” 2 1 2 • Complementary Here, the red box tells us the whole angle is 90o 1 So we write the equation: 2 m1 m2 90 • Supplementary Here, since the whole angle is straight, we know it is 180o 1 So we write the equation: 2 m1 m2 180 Angle Bisector A line, segment, or ray that cuts an angle into 2 equal pieces. 60o 60o Linear Pairs Two adjacent angles are a linear pair if their noncommon sides form a line. 5 6 5 and 6 are a linear pair. Linear pairs are always supplementary 5 6 180 o Finding Measures of Complements and Supplements Find the angle measure. Given that A is a complement of C and m A = 47˚, find mC. SOLUTION mC = 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 mC. Given that P is a supplement of R and mR = 36˚, find mP. SOLUTION mC = 90˚ – m A mP = 180˚ – mR = 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˚. Simplifying gives 6(m W) = 90˚, Divide both sides by 6 to get m W = 15˚. Finding Angle Measures Solve for x and y. Then find the angle measure. ( 3x + 5)˚ D • E ( x + 15)˚ ( 4y – 15)˚ • B A• ( y + 20)˚ • C SOLUTION Use the fact that the sum of the measures of angles that form a Use substitution to find the angle measures (x = 40, y = 35). linear pair is 180˚. m AED = ( 3 x + 15)˚ = (3 • 40 + 5)˚ = 125˚ m AED + m DEB = 180° m AEC + mCEB = 180° m + 15)˚ = (40 + 15)˚ = 55˚ ( 3x + DEB 5)˚ + = ( x(+x 15)˚ = 180° ( y + 20)˚ + ( 4y – 15)˚ = 180° m AEC = 4x ( y ++ 20 20)˚ = (35 + 20)˚ = 55˚ = 180 5y + 5 = 180 m CEB = ( 4 y –4x15)˚ = (4 • 35 – 15)˚ = 125˚ = 160 5y = 175 x = 40 y = the 35 vertical So, the angle measures are 125˚, 55˚, 55˚, and 125˚. Because angles are congruent, the result is reasonable.