Isosceles, Equilateral, and Right Triangles UNIT 4 LESSON 5

Transcription

1 Isosceles, Equilateral, and Right Triangles UNIT 4 LESSON 5

2 Objectives: Use properties of isosceles and equilateral triangles Use properties of right triangles

3 ssignment: pp #1-26, 29-32, 33, 39

4 Using properties of Isosceles Triangles triangle is an isosceles if it has at least two congruent sides. If it has exactly two congruent sides, then they are the legs of the triangle and the noncongruent side is the base. The two angles adjacent to the base are the base angles. The angle opposite the base is the vertex angle. leg vertex angle leg base angles base

5 Investigating Isosceles Triangles 1. Use a straight edge and a compass to construct an acute isosceles triangle. Then fold the triangle along a line that bisects the vertex angle as shown. 2. Repeat the procedure for an obtuse isosceles triangle. 3. What observations can you make about the base angles of an isosceles triangle? Write your observations as a conjecture (what did you observe?).

6 What did you discover? In the activity, you may have discovered the ase ngles Theorem, which is proved in Example 1 which follows this slide. The converse of this theorem is also true.

7 Theorems ase ngles Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent. If, then.

8 Theorems onverse of the ase ngles Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent. If, then.

9 Ex. 1: Proof of the ase ngles Theorem Given:, Prove: Paragraph proof: Draw the bisector of. y construction, D D. You are given that. lso, D D by the Reflexive property of ongruence. Use the SS ongruence postulate to conclude that D D. ecause PT, it follows that.

10 Remember: n EQUILTERL triangle is a special type of isosceles triangle. The corollaries below state that a triangle is EQUILTERL if and only if it is EQUINGULR. orollary to theorem If a triangle is equilateral, then it is equiangular. orollary to theorem If a triangle is equiangular, then it is equilateral.

11 Ex. 2: Using Equilateral and Isosceles Triangles a. Find the value of x b. Find the value of y Solution a: How many total degrees in a triangle? This is an equilateral triangle which means that all three angles are the same. 3x = 180 Triangle Sum Theorem. X = 60 x y

12 Ex. 2: Using Equilateral and Isosceles Triangles a. Find the value of x b. Find the value of y Solution b: How many total degrees in a line? The triangle has base angles of y which are equal. (ase ngles Theorem). The other base angle has the same measure. The vertex angle forms a linear pair with a 60 angle, so its measure is y = 180 (Triangle Sum Theorem) 2y = 60 (Solve for y) y = 30 x 60 y

13 Using Properties of Right Triangles You have learned four ways to prove that triangles are congruent. Side-Side-Side (SSS) ongruence Postulate Side-ngle-Side (SS) ongruence Postulate ngle-side-ngle (S) ongruence Postulate ngle-ngle-side (S) ongruence Theorem The Hypotenuse-Leg ongruence Theorem on the next slide can be used to prove that two RIGHT triangles are congruent.

14 Hypotenuse-Leg (HL) ongruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. If EF and DF, then DEF. E F D

15 Ex. 3: Proving Right Triangles ongruent The television antenna is perpendicular to the plane containing points,, D, and E. Each of the stays running from the top of the antenna to,, and D uses the same length of cable. Prove that E, E, and ED are congruent. Given: E E, E E, E ED, D. Prove E E ED D E

16 Given: E E, E E, E ED, D. Prove E E ED D E Paragraph Proof: You are given that E E, E E, which implies that E and E are right angles. y definition, E and E are right triangles. You are given that the hypotenuses of these two triangles, and, are congruent. lso, E is a leg for both triangles and E E by the Reflexive Property of ongruence. Thus, by the Hypotenuse-Leg ongruence Theorem, E E. Similar reasoning can be used to prove that E ED. So, by the Transitive Property of ongruent Triangles, E E ED.

17 Proof: Given: Prove: Draw the angle bisector of.

18 Proof: Given: Prove: Statements: D is bisector of 3. D D 4. D D 5. E E 6. D D 7. Reasons: 1. Given

19 Proof: Given: Prove: Statements: D is bisector of 3. D D 4. D D 5. E E 6. D D 7. Reasons: 1. Given 2. y construction

20 Proof: Given: Prove: Statements: D is bisector of 3. D D 4. D D 5. E E 6. D D 7. Reasons: 1. Given 2. y construction 3. Definition isector

21 Proof: Given: Prove: Statements: D is bisector of 3. D D 4. D D 5. E E 6. D D 7. Reasons: 1. Given 2. y construction 3. Definition isector 4. Third ngles Theorem

22 Proof: Given: Prove: Statements: D is bisector of 3. D D 4. D D 5. E E 6. D D 7. Reasons: 1. Given 2. y construction 3. Definition isector 4. Third ngles Theorem 5. Reflexive Property

23 Proof: Given: Prove: Statements: D is bisector of 3. D D 4. D D 5. E E 6. D D 7. Reasons: 1. Given 2. y construction 3. Definition isector 4. Third ngles Theorem 5. Reflexive Property 6. S ongruence Postulate

24 Proof: Given: Prove: Statements: Reasons: D is bisector of 3. D D 4. D D 5. E E 6. D D Given 2. y construction 3. Definition isector 4. Third ngles Theorem 5. Reflexive Property 6. S ongruence Postulate 7. PT