Warm-Up Based on upper. Based on lower boundary of 1. m 1 m 2 m 3 m What do you notice about these angles?
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1 Warm-Up Metalbro is a construction company involved with building a new skyscraper in ubai. The diagram below is a rough sketch of a crane that Metalbro workers are using to build the skyscraper. The vertical line represents the support tower and the other line represents the boom. For safety reasons, the boom cannot be more than 15º beyond the horizontal in either direction. horizontal line forms a 90º angle with the support tower. straight line forms a 180º angle What are the safety requirements for m 1? 2. What are the safety requirements for m 2? 3. What are the safety requirements for m 3? 4. What are the safety requirements for m 4? 5. Use your findings to fill in the table. m 1 m 2 m 3 m 4 ased on lower boundary of 1 ased on upper boundary of 1 6. What do you notice about these angles?
2 Straight angles are angles with rays in opposite directions in other words, straight angles are straight lines. Straight angle Not a straight angle is a straight angle. Points,, and lie on the same line. P Q PQR is not a straight angle. Points P, Q, and R do not lie on the same line. R djacent angles are angles that lie in the same plane and share a vertex and a common side. They have no common interior points. Nonadjacent angles have no common vertex or common side, or have shared interior points. djacent angles Nonadjacent angles P F Q S R is adjacent to. They share vertex and. and have no common interior points. is not adjacent to F. They do not have a common vertex. PQS is not adjacent to PQR. They share common interior points within PQS.
3 Linear pairs are pairs of adjacent angles whose non-shared sides form a straight angle. Linear pair Not a linear pair F and are a linear pair. They are adjacent angles with non-shared sides, creating a straight angle. and F are not a linear pair. They are not adjacent angles. Vertical angles are nonadjacent angles formed by two pairs of opposite rays. Vertical ngles Vertical angles are congruent. Vertical angles Not vertical angles and are vertical angles. and are vertical angles. and are not vertical angles. and are not opposite rays. They do not form one straight line.
4 Postulate ngle ddition Postulate If is in the interior of, then m + m = m. If m + m = m, then is in the interior of. Informally, the ngle ddition Postulate means that the measure of the larger angle is made up of the sum of the two smaller angles inside it. Supplementary angles are two angles whose sum is 180º. Supplementary angles can form a linear pair or be nonadjacent. In the following diagram, the angles form a linear pair. m + m = 180 The next diagram shows a pair of supplementary angles that are nonadjacent. m PQR + m TUV = 180 P T Q 25º R V U 155º Supplement If two angles form a linear pair, then they are supplementary.
5 ngles have the same congruence properties that segments do. ongruence of angles is reflexive, symmetric, and transitive. Reflexive Property: 1 1 Symmetric Property: If 1 2, then 2 1. Transitive Property: If 1 2 and 2 3, then 1 3. ngles supplementary to the same angle or to congruent angles are congruent. If m 1+ m 2= 180 and m 2+ m 3= 180, then 1 3. Perpendicular lines form four adjacent and congruent right angles. If two congruent angles form a linear pair, then they are right angles. If two angles are congruent and supplementary, then each angle is a right angle. The symbol for indicating perpendicular lines in a diagram is a box at one of the right angles, as shown below. Q P R S The symbol for writing perpendicular lines is, and is read as is perpendicular to.
6 Remember that perpendicular bisectors are lines that intersect a segment at its midpoint at a right angle; they are perpendicular to the segment. ny point along the perpendicular bisector is equidistant from the endpoints of the segment that it bisects. Perpendicular isector If a point lies on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. If is the perpendicular bisector of, then =. If =, then is the perpendicular bisector of. omplementary angles are two angles whose sum is 90º. omplementary angles can form a right angle or be nonadjacent. The following diagram shows a pair of nonadjacent complementary angles.
7 m + m = 90 55º F 35º The next diagram shows a pair of adjacent complementary angles labeled with numbers. m 1+ m 2= 90 P S Q 1 2 R omplement If the non-shared sides of two adjacent angles form a right angle, then the angles are complementary. ngles complementary to the same angle or to congruent angles are congruent.
8 Guided Practice xample 1 Look at the following diagram. List pairs of supplementary angles, pairs of vertical angles, and a pair of opposite rays F xample 3 In the diagram below, and are intersecting lines. If m 1= 3x+ 14 and m 2= 9x+ 22, find m 3 and m
9 xample 5 In the diagram below, is the perpendicular bisector of. If = 4x 1 and = x + 11, what are the values of and?
10 Problem-ased Task 1.8.1: utting Kitchen Tiles Maresol is retiling the backsplash over her kitchen stove, and has to cut square ceramic tiles into 4 congruent triangles to create the pattern she wants. If she doesn t cut each square into perfectly equal triangles, the tiles won t fit together properly. efore she cuts the first tile, she uses a pencil to draw two segments on the tile. The segments form perpendicular bisectors. Use what you know about triangle congruency and perpendicular bisectors to prove that the 4 triangles are congruent.
11 Practice 1.8.1: Proving the Vertical ngles Use the following diagram to solve problems List two pairs of adjacent angles and two pairs of nonadjacent angles. 2. List two pairs of supplementary angles. Write a statement about those angles using the Supplement. 3. List a pair of vertical angles. Write a statement about those angles using the Vertical ngles. 4. List a pair of complementary angles. Write a statement about those angles using the omplement. continu
12 In the diagram that follows, and intersect. Use this information to solve for the measures of the unknown angles in problems 5 and 6. Show and justify your work Find m 4 if m 1= 3x+ 4 and m 2= 2x Find m 1 if m 1= 13x+ 7 and m 3= 7x+ 49.
13 Use the diagram that follows to solve problems 7 and Find m 1 given the following: 3 and 4 are complementary, 2 3, m 2= 6x+ 24, and m 4= 5x. 8. Find m 1 if 2 and 3 are complementary, m 1= 4x 23, and m 4= x+ 38.
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