Warm Up 1 ft. Find the area of the green region. Assume all angles are right angles. 2 ft. 1 ft. 2 ft. 4 ft. 3 ft. 1 ft

Download Warm Up 1 ft. Find the area of the green region. Assume all angles are right angles. 2 ft. 1 ft. 2 ft. 4 ft. 3 ft. 1 ft

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<ul><li> Slide 1 </li> <li> Slide 2 </li> <li> Warm Up 1 ft. Find the area of the green region. Assume all angles are right angles. 2 ft. 1 ft. 2 ft. 4 ft. 3 ft. 1 ft. </li> <li> Slide 3 </li> <li> Sections 3.6 and 3.7 </li> <li> Slide 4 </li> <li> Congruent Angles Definition: they have the same degree measure. Symbols: angle A congruent angle B if and only if m a and m b Picture: 30 B 30 A </li> <li> Slide 5 </li> <li> The first relationship we are going to talk about Definition: Two angles are vertical angles if their sides form two pairs of opposite rays Angles 1 and 2 are vertical angles 1 2 34 Angles 3 and 4 are also vertical angles Vertical angles are always congruent. </li> <li> Slide 6 </li> <li> Theorem 3.1-Vertical Angle Thm Definition: vertical angles are congruent Picture: Symbols: Angles 1 and 3 are congruent Angles 2 and 4 are congruent 2 4 1 3 </li> <li> Slide 7 </li> <li> 5y 50 4y 10 What type of angles are these? 5y 50 = 4y 10 y = 40 Plug y back into our angle equations and we get What is the measure of the angle? </li> <li> Slide 8 </li> <li> Find the value of x in each figure 1. 2. 3.4. 130 x 5x 25 x40 (x 10) 125 </li> <li> Slide 9 </li> <li> Theorem 3-2: If two angles are congruent, then their complements (90 ) are congruent. Picture: 60 60 The measure of angles complementary to A and B is 30. </li> <li> Slide 10 </li> <li> 1 2 3 4 5 Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example 1: ADJACENT </li> <li> Slide 11 </li> <li> 1 2 3 4 5 Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example 2: VERTICAL </li> <li> Slide 12 </li> <li> 1 2 3 4 5 Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example 3: ADJACENT, COMPLEMENTARY </li> <li> Slide 13 </li> <li> Theorem 3-3: If two angles are congruent, then their supplements (180 ) are congruent. 70 110 110 70 Angle 1 is congruent to angle 4 (measure of 70) </li> <li> Slide 14 </li> <li> Theorem 3-4: If two angles are complementary to the same angle, then they are congruent. Angle 3 is complementary to angle 4, angle 5 is complementary to angle 4. Angle 3 is congruent to angle 5. 3 4 5 </li> <li> Slide 15 </li> <li> Theorem 3.5: If two angles are supplementary to the same angle, then they are congruent. Angle 1 is supplementary to angle 2. Angle 3 is supplementary to angle 2. Angle 1 congruent to angle 3. 1 3 2 </li> <li> Slide 16 </li> <li> 1 2 3 4 5 Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Example 4: ADJACENT, SUPPLEMENTARY, LINEAR PAIR </li> <li> Slide 17 </li> <li> Find x, y, and z. Example 5: x = 129, y = 51, z = 129 </li> <li> Slide 18 </li> <li> Find x. Example 6: X = 8 L PA T O </li> <li> Slide 19 </li> <li> Find Example 7: 155 L PA T O Since we have already found the value of x, all we need to do now is to plug it in for LAT. </li> <li> Slide 20 </li> <li> Theorem 3-6: If two angles are congruent and supplementary, then each is a right angle. 1 2 </li> <li> Slide 21 </li> <li> Theorem 3-7: All right angles are congruent. These angles are congruent </li> <li> Slide 22 </li> <li> Moving right along Section 3-7- Perpendicular Lines This is the last section of the chapter! YEAH. </li> <li> Slide 23 </li> <li> Lines that intersect to form four right angles are perpendicular lines. m l 12 4 3 Symbol: m l is read as m perpendicular to l </li> <li> Slide 24 </li> <li> Clipper- Flying Cloud Ship The main mast and the frame for the sails are examples of perpendicular line segments. The main mast is perpendicular to the sail frame, the likewise, the frame for the sail is perpendicular to the main mast. </li> <li> Slide 25 </li> <li> Flying Cloud One common nineteenth century ship was the clipper. This ship, which had many sails, was designed for speed. In fact, it was named a clipper because of the way it clipped off the miles. </li> <li> Slide 26 </li> <li> Theorem 3-8: If two angles are perpendicular, then they form four right angles. Each angle is 90 degrees. Each angle is congruent. </li> <li> Slide 27 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 28 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 29 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 30 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 31 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 32 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 33 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 34 </li> <li> 1 2 3 4 5 6 B G V F A C E 8 </li> <li> Slide 35 </li> <li> Theorem 3.9: If a line m is in a plane and point T is a point on m, then there exists exactly one line in that plane that is perpendicular to m at T. m TT </li> <li> Slide 36 </li> <li> Slide 37 </li> <li> Homework: workbook page 17 and 18 ALL </li> <li> Slide 38 </li> </ul>

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