TOPIC: Isosceles Triangle

Isosceles Triangle Ref/Info from book (pls read) Explore The Isosceles Triangle Recall the parts of the isosceles triangle. O vertex angle leg -leg base angle base angle R SX base 1. Cut out an isosceles triangle and name it AROX. 2. Fold the triangle along its altitude. 3. Compare the base angles of the triangle. Are they congruent? 4. Unfold the triangle. The crease made two right triangles. Are the two triangles congruent? Why do you say so?The figure below shows the unfolded triangle. RO = XO . Why? ON = ON. Why? N What postulate guarantees that ARON = AXON? Thus, what theorem guarantees that the base angles ZR and ZX are congruent? The results of this activity lead us to this theorem. THEOREM 6-1 Isosceles Triangle Theorem The angles opposite the congruent sides of an isosceles triangle are congruent. The details of the proof are left to the student as Exercise #7. Suppose you are told that the two angles of a triangle are congruent. Can you conclude that the opposite sides of each of the congruent angles are congruent? Suppose you have AABC with ZA = ZC. Let's draw the perpendicular bisector from B to AC at point D. Angles BDA and BDC are right angles. Why? Therefore, LBDA = ZBDC. Now, AD = CD . Why? Think What postulate will show that AABD = ACBD? If AABD = ACBD, can we conclude that AB = CB? About This How does the This leads us to the next theorem. Isosceles Triangle Theorem make the job of proving congruence THEOREM 6-2 between two If two angles of a triangle are congruent, then the sides opposite them isosceles triangles easier? are congruent. The proof of this theorem is given as Exercise #13. We will discuss the next theorem since it makes use of the Isosceles Triangle Theorem. THEOREM 6-3 An equilateral triangle is also equiangular.O Given: AJOY with JO = OY = JY Prove: LJ = ZO = LY Proof: Statements Reasons 1 . OY = JY 1. Given 2. U= ZO 2. Isosceles Triangle Theorem 3. JO = JY 3. Given 4. LO = LY 4. Isosceles Triangle Theorem 5 . L= LO= LY 5. Transitivity (2, 4) Is the converse of this theorem also true? Meaning, is an equiangular triangle also equilateral? THEOREM 6-4 An equiangular triangle is also equilateral. Proof of this theorem is given as Exercise #14. Establishing congruence between two triangles can be useful when you want to prove congruence between corresponding segments or angles. The statement that follows is a takeoff from the definition of congruent triangles. It is most often used in proving congruence between corresponding parts of the triangles. CPCTC Corresponding parts of congruent triangles are congruent. EXAMPLE 1 A pole is supported by two congruent wire braces. Each brace is two meters away from the foot of the pole in opposite directions. Prove that the angles made by the braces with the pole are congruent. Let us label the parts of the two triangles as shown at the right. In the diagram, ED represents the pole, BE and RE the braces, and BD and RD the respective distances of each brace from the pole. The formal proof shows how the CP he CPCTC is used. Given: ABER is isosceles; BD = RD Prove: ZBED = LRED B 2m D 2m RSOLUTION: Proof: Statements 0 0 MINORHT 1. BD = RD Reasons 1. Given 2. BE = RE 3. LB = LR 2. Definition of Isosceles Triangles 4. ABED = ARED 3. Isosceles Triangle Theorem . ZBED = ZRED 4. SAS Postulate (1, 2, 3) 5. CPCTC 7.4.2 TRIANGLE INEQUALITIES The next theorems are on triangle inequalities. gris go tobite owls In the previous chapter, we used inductive reasoning to arrive at the theorem. This time, we will prove the theorem using the indirect method. In the indirect method of proving, we assume that what we want to prove is false. Then we try to show that there is a contradiction. This proves indirectly that what we assumed to be false is actually true. THEOREM 6-5 Angle-Side Inequality Theorem In a triangle, if one angle is larger than the other angle, the side opposite the larger angle is the longer side. EXAMPLE 2 Prove the Angle-Side Inequality Theorem: Given: ARED with mZE > mZD Prove: RD > RE SOLUTION: Proof: One and only one of these relations is true: RD > RE, RD = RE or RD RE is false. We have to prove that RD = RE or RD mZD. Hence, RD = RE. Case 2: RD mZD. It is a contradiction. Hence, RD RE.THEOREM 6-6 Side-Angle Inequality Theorem In a triangle, if one side is longer than the other side, the angle opposite the longer side is the larger angle. The proof of the Side-Angle Inequality Theorem is given as Exercise #6. Let us explore another triangle inequality theorem. - - . ACTIVITY NO. 1 . - - - - - - More Triangle Inequalities Steps: 1. Take two sticks or straws of unequal lengths. Place them together to form an angle. Imagine that the sticks are connected at A by a hinge with the other ends B and C joined by a rubber band. 2. As the hinge is opened wider, the rubber band ought to be stretched. A 3. What can you say about the length of BC as m/BAC increases? The result of this activity leads to the Hinge Theorem. THEOREM 6-7 The Hinge Theorem If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.Activity, pls answer vocabulary and Concepts ly a valid conclusion in (a) and the corresponding reason in (b) for each of the following. 1. In ABRW. BR = BW, (a) (b) 2. In AOMG, LG = ZM, (a) ( b ) Practice and Application Refer to the given figure to answer 1-5. 1. If AC = BC, name two congruent angles. 2. If 27 = 28, name two congruent segments. C 3. If WB = WC, name two congruent angles. 4. If 23 = 48, name two congruent segments. 5. If AB = BC, name two congruent angles. W II. Find the value of x. 8. 9 X 6. 7. 80 xo 2x - 3 11 1750Comprehension Check Fill in the blanks with the correct word or phrase to complete each statement. 1. In a triangle, the angle opposite the longest side is the angle. 2. In a triangle, the side opposite the smallest angle is the side. 3. Corresponding parts of are congruent. 4 . An equilateral triangle is also a/an triangle. 5. The base angles of an isosceles triangle are Mathematical Reasoning Complete the proof of the following by providing the reasons for the statements. 6. Prove the Side-Angle Inequality Theorem: In a triangle, if one side is longer than the other side, the angle opposite the longer side is the larger angle. Given: AMNO with MO > MN N bos JA lo za Prove: mZN > mZO MProof. Statements Reasons 1. AMNO with MO > MN 1. 2. On MO, construct P such that 2. MN = MP ; Draw NP. 3. 41= 42 3. 4. m/2 = m mzo 5 . 6. m21>mZo 6. 7. m/N= m m /1 8. 9 . m /N> mZo 9 . 7. Prove the Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite these sides are congruent. Given: AMIN with MI = IN Prove: LM = ZN Proof: M Statements Reasons 1. MI = IN ; IN = MI 1. 2. LI= LI 2. 3. AIMN = AINM 3. 4. LM = ZN 4. 8. Given: B is the midpoint of AC and DE. A Prove: AD = EC 8 D C Proof: Statements Reasons 1. B is the midpoint of AC and DE. 1. 2. AB = BC 2. 3. DB = BE 3. 4. ZABD and ZCBE are vertical angles. 4. 5. LABD = LCBE 5 . 6. AABD = ACBE 6 .Given: ADEF is isosceles with vertex at E. EG bisects LDEF D Prove: ADEG = AFEG quitvled molder G E TI 10. Given: AABC is isosceles with vertex at B. BD J AC Prove: AABD = ACBD A D 11. Given: AABC is isosceles with vertex at B. B BD bisects LABC a. Prove: AABD = ACBD b. Prove: D is the midpoint of AC. A D 12. A kite is formed as shown in the diagram. Given: AABC and ACDA are isosceles triangles. BD and AC are diagonals with vertices B and D, respectively. a. Prove that AABD = ACBD. b. Prove that ZA = ZC. 13. If two angles of a triangle are congruent, then the sides opposite them are congruent. 14. An equiangular triangle is also equilateral.Problem Solving 15. A 4m X 8m rectangular billboard on top of a building is braced by two wires. The other end of each wire is 4 meters from the base of the billboard. Prove that the two triangles formed by the wire with the sides of the rectangles and the top of the building are congruent. Challenge Yourself 16. Given: AE = DE; EB = EC; AB = CD; Prove: AAEC = ADEB A D B C 17. Given: AC = EC; ZFAE = ZFEA; D C B AB = ED ; Prove: ACAD = ACEB A E DA bris 08 18. Prove: The sum of the lengths of any two sides of a triangle is greater than the length of the third side