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Angle Relationships Warm Up Identify the type of angle. 1. 70° acute 2. 90° right 3. 140° obtuse 4. 180° straight Today we will learn to understand new relationships of angles. Vocabulary congruent vertical angles adjacent angles complementary angles supplementary angles Angle Relationships Complementary angles are two angles whose measures have a sum of 90°. 65° + 25° = 90° LMN and NMP are complementary. L N 65° 25° M Course 1 P You can use what you know about right angles to identify an unknown angle measure. Find each unknown angle measure. The angles are complementary. We know one is 71. 71° + a = –71° a= We know the sum must be 90. 90° –71° 19° The sum of the measures is 90°. a 71° Find the unknown angle measure. The angles are complementary. 65° + d = 90° –65° –65° d = 25° The sum of the measures is 90°. d 65° Supplementary angles are two angles whose measures have a sum of 180°. 65° + 115° = 180° GHK and KHJ are supplementary. K 65° G Connecting the two makes a semicircle on a line. 115° H J We can use what we know about straight angles to find a missing angle measure. Find each unknown angle measure. The angles are supplementary. The sum of the measures is 180°. 125° + b = 180° –125° –125° b= 55° b 125° Find the unknown angle measure. The angles are supplementary. 145° + s = 180° –145° –145° s= 35° The sum of the measures is 180°. 145° s When angles have the same measure, they are said to be congruent. Use what we know about straight angles to find a missing angle measure. JKL and MKN are congruent. We want to find the measures of two congruent angles.We will let n stand for the measure of one. Same measure, same variable…. M L n J 80° K n1 N 2n + 80° = 180° The sum of the measures is –80° –80° 180°. 2n = 100° So…both = 50° n = 50° and n1 = 50° Each angle measures half of 100°. Find each unknown angle measure. C ABC and DBE are congruent. D 50° n A n1 B E 2n + 50° = 180° The sum of the measures is –50° –50° 180°. 2n = 130° n = 65° and n1 = 65° Each angle measures half of 130°. M N 20° P 160° R 160° 20° Q Vertical angles are formed opposite each other when two lines intersect. Vertical angles have the same measure, so they are always congruent. MRP and NRQ are vertical angles. MRN and PRQ are vertical angles Identifying Types of Angle Pairs: Extra Info Identify the type of each angle pair shown. 5 6 Imagine two triangles pointing towards each other. These would be inside the vertical angles. 5 and 6 are opposite each other and are formed by two intersecting lines. They are vertical angles. These two angles are also vertical angles. Find each unknown angle measure. The angles are vertical angles. (given) c = 82° Vertical angles are congruent. c 82° d WHY????? If we know that angle C = 82 degrees, what is the measure of angle d? 180 = 82 + d 180 = 82 + 92 Let’s use the previous example in your notes to discuss adjacent angles. Adjacent angles are side by side and have a common vertex and ray. Adjacent angles may or may not be congruent. M N 20° P 160° R 160° 20° Q MRN and NRQ are adjacent angles. They share vertex R and RN. How else could these two angles be classified? They are also supplementary. NRQ and QRP are adjacent angles. They share vertex R and RQ. They are also ??????? They are also supplementary. These two angles are adjacent. They are not supplementary or complementary. 7 and 8 are side by side and have a common vertex and ray. 7 8 They are adjacent angles. 3 4 3 and 4 are side by side and have a common vertex and ray. They are adjacent angles. They are also supplementary. Lesson Quiz, Part I Give the complement of each angle. 1. 70° What do I need 2. n=20°to = 90? 42° n=48° What do I need to = 90? Give the supplement of each angle. 3. 120° n=60 4. 17° 163° ° 5. Identify the type of angle pair shown. adjacent Why are they NOT supplementary? What do I need to = 180? Time Check Oh yes, there is more. Not today. We have several new angle relationships and the algebraic application for finding missing angles. Angle Relationships II Vocabulary • • • • Corresponding Angles Alternate Exterior Angles Alternate Interior Angles Transversal More Angle Relationships In this picture, we have two parallel lines intersected by another line. This intersecting line is called a transversal. When a transversal intersects two parallel lines, we create several new angle relationships. Corresponding angles are angles that lie on the same side of the transversal and the same side of the (respective) parallel line. I call this the Walgreens’ Property. We’ll get to your notes in a minute. This could also be called the Starbucks’ Property….. or the Dollar General Property….. Now let’s identify the corresponding angles….The letters in your notes represent the different angles created by the parallel lines and the transversal. a h e c f b g d The corresponding angles are: a and c, e and h, b and d, f and g Corresponding angles are congruent . Alternate interior angles lie inside the region of the parallel lines, on alternate sides of the transversal. h c g d Alternate Interior Angles: c and g, d and h Alternate interior angles are congruent to each other. Alternate exterior angles are angles that lie on opposite sides of the transversal, outside of the parallel lines. a e f Alternate exterior angles are congruent. A and f, b and e are alternate exterior angles….. b If we know the measure of one angle, we can find the rest. We will find them in this order: a (given), g, f, c, e, d, h, b a has a measure of 135o g is also 135o supplementary f is 135o c is 135o vertical supplementary 45o h alt. interior corresponding corresponding vertical alt. interior b 45o d f 135o alt exterior e must be d is 45o h is 45o b is 45o vertical supplementary corresponding 45o corresponding alternate ext. 45o g 135o vertical 135oc 45o e supplementary 135o a We will use the letter on the diagram to represent the angles created by the intersection of these lines. a has a measure of 80o 80o a b 100o g 80o 100o h 80o c 100o f o d 100 e 80o Geometry + Algebra??? 6(8) + 60 = 48 + 60 = 108° (6x + 60)° 108° 72°(9x)° 6x + 60 + 9x = 180 15x + 60 = 180 -60 -60 15x = 120 x=8 9x = 9(8) 9x = 72° 1 72° Corresponding angles 2 These two lines are parallel. 108° Alternate exterior angles….and? 52° 1 15x – 7 + 5x + 7 = 180 20x = 180 (15x – 7)° 128° Supplementary x = 9° 52° 15x - 7 = 15(9) - 7 = 135 - 7 = 128° Vertical (5x + 7)° 5x + 7 = 5(9) + 7 = 45 + 7 = 52° 2 128° Corresponding Did we learn more relationships of angles?