Warm-Up 3/30/ What is the measure of angle ABC.

Transcription

1 enchmark #3 Review

2 Warm-Up 3/30/ What is the measure of angle.

3 Warm-Up 3/31/ Five exterior angles of a convex hexagon have measure 74, 84, 42, 13, 26. What is the measure of the 6 th exterior angle?

4 Warm-Up 4/1/15 1. a. 5, 7, 9 b. 16, 30, What is the area of the largest square in the figure below?

5 Right Triangles: In a right triangle, the side opposite the right angle is the longest side, called the hypotenuse. The other two sides are the legs of a right triangle. Theorem 7.1 Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2

6 The onverse of the Pythagorean Theorem is used to determine if a triangle is a right triangle, acute triangle, or obtuse triangle. If c 2 = a 2 + b 2, then the triangle is a right triangle. If c 2 > a 2 + b 2, then the triangle is an obtuse triangle. If c 2 < a 2 + b 2, then the triangle is an acute triangle.

7 Special Right Triangle In a triangle , the hypotenuse is 2 times as long as a leg. Example: Leg X 45 Hypotenuse X 2 5 cm cm cm Leg Special Right Triangles X

8 Special Right Triangle In a triangle , the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. 3 Longer Leg X 3 30 Hypotenuse 2X 5 3 cm Example: cm Special Right Triangles X 60 Shorter Leg 60 5 cm

9 If the reference angle is then dj Opp Hyp then Opp dj Hyp Hyp Hyp Opp Opp dj dj 9 Right Triangle Trigonometry

10 Definitions of Trig Ratios Opp Sin Hyp dj os Hyp hypotenuse hypotenuse opposite Tan Opp adjacent dj

11 SOH- H -TO Opp dj Opp Sin os Tan Hyp Hyp dj

12 SOHHTO sin opp hyp cos tan adj hyp opp adj

13 Solving Trigonometric Equations There are only three possibilities for the placement of the variable x. sin = Opp x sin = x Hyp Sin X = Opp Hyp 12 cm x 25 sin 25 = 12 x sin 25 1 = 12 x x = 12 sin 25 x sin 25 = sin 25 = 1 x cm 25 x 12 x = (12) (sin 25) 12 cm sin x = 25 cm x x = sin (12/25) x = x = 28.4 cm x = 5.04 cm Note you are looking for an angle here!

14 H 7.3 D When you write a proportion comparing the legs lengths of D and D, you can see that D is the geometric mean of D and D. D D Shorter leg of D. D D = D D Longer leg of D. Shorter leg of D Longer leg of D.

15 Geometric Mean Theorems Theorem 7.6: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments Theorem 7.7: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. D D D = D D = = D D

16 What does that mean? 2 x y x = x y = y 2 18 = x 2 18 = x 7 y = y = x 14 = y = x 14 = y

17 If a convex polygon has n sides, then the sum of the measure of the interior angles is (n 2)(180 )

18 If a regular convex polygon has n sides, then the measure of one of the interior angles is ( n 2) 180 n

19 Parallelogram Definition: quadrilateral whose opposite sides are parallel. D and D Symbol: a smaller version of a parallelogram Naming: parallelogram is named using all four vertices. You can start from any one vertex, but you must continue in a clockwise or counterclockwise direction. For example, the figure above can be either D or D. D

20 Properties of Parallelogram 1. oth pairs of opposite sides are congruent. D and D 2. oth pairs of opposite angles are congruent. and D 3. onsecutive angles are supplementary. mm180 and mmd180 mm 180 and mmd Diagonals bisect each other but are not congruent P is the midpoint of and D. D P P P P PD

21 Theorems D is a parallelogram. Theorem 8.7: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. D

22 Theorems D is a parallelogram. Theorem 8.8: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. D

23 Theorems D is a parallelogram. Theorem 8.9: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. D

24 Theorems D is a parallelogram. Theorem 8.10: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. D

25 Properties of Special Parallelograms In this lesson, you will study three special types of parallelograms: rhombuses, rectangles and squares. rectangle is a parallelogram with four right angles. rhombus is a parallelogram with four congruent sides square is a parallelogram with four congruent sides and four right angles.

26 Take note: orollaries about special quadrilaterals: Rhombus orollary: quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle orollary: quadrilateral is a rectangle if and only if it has four right angles. Square orollary: quadrilateral is a square if and only if it is a rhombus and a rectangle.

27 Venn Diagram shows relationships Each shape has the properties of every group that it belongs to. For instance, a square is a rectangle, a rhombus and a parallelogram; so it has all of the properties of those shapes. parallelograms rhombuses squares rectangles