Proving Tпangles Similar

Practice **Proving Triangles Similar** Determine whether the triangles are similar. If so, write a similarity statement and name the postulate or theorem   you used. If not, explain. 1. [Image of two triangles: AEB and DEC] ΔABE ~ ΔDCE by the AA Postulate 2. [Image of two triangles: KLM and NOP] Not similar; only one side and one angle are congruent. 3. [Image of two triangles: LMN and OPQ] ΔLMN ~ ΔOPQ by the AA Postulate 4. [Image of two triangles: ABC and DEF] Not similar; only one angle and one side are congruent. 5. [Image of two triangles: TUV and WXY] ΔTUV ~ ΔWXY by the SAS Theorem 6. [Image of two triangles: MNP and OQR] ΔMNP ~ ΔOQR by the SSS Theorem 7. Given: RM || SN, RM ⊥ MS, SN ⊥ NT Prove: ΔRSM ~ ΔSTN [Image of two right triangles: RSM and STN] Statements: 1) RM || SN, RM ⊥ MS, SN ⊥ NT 2) ∠MRS ≅ ∠NST 3) ∠RMS ≅ ∠NTS 4) ΔRSM ~ ΔSTN Reasons: 1) Given 2) Corresponding angles 3) Perpendicular lines form right angles 4) All right angles are congruent 5) AA Postulate 8. Given: A bisects IK, C bisects KL, B bisects IL. Prove: ΔIKL ~ ΔCBA [Image of two triangles: IKL and CBA] It is given that A, C, and B are the midpoints of IK, KL, and IL. Therefore, according to the Midsegment Theorem, AC is half the length of IK, BC is half the length of KL, and AB is half the length of IL. It follows then that ΔIKL ~ ΔCBA by the SSS Theorem. 9. A 1.4-m tall child is standing next to a flagpole. The child's shadow is 1.2 m long. At the same time, the shadow of the flagpole is 7.5 m long. How tall is the flagpole?   8.75 m Practice (continued) **Proving Triangles Similar: Then Find the Value of x** 10. [Image of two triangles: OPN and LMN] ΔOPN ~ ΔLMN because if two sides of a triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are   similar by the SAS Theorem. OP/LM = NP/MN = ON/LN 10/20 = 9/18 = 5/12 So, x = 12 11. [Image of two triangles: ABC and ADE] ΔABE ~ ΔACD by the AA Postulate. AB/AD = BC/DE 3x/4x-1 = 15/18 Solving for x, we get x = 7. 12. A stick 2 m long is placed vertically at point B. The top of the stick is in line with the top of a tree as seen from point A, which is 3 m from the stick and 30 m from the tree. How tall is the tree? 20 m   [Image of a stick and a tree with similar triangles] 13. Thales was an ancient philosopher familiar with similar triangles. One story about him says that he found the height of a pyramid by measuring its shadow and his own   shadow at the same time. If the person is 5-ft tall, what is the height of the pyramid in the drawing? 265   ft [Image of a person and a pyramid with their shadows] **Identify the similar triangles in each figure. Explain.** 14. [Image of a triangle with various line segments] ΔADB ~ ΔABC (AA ~); ΔADC ~ ΔABC (AA ~); ΔADB ~ ΔADC (Trans. Prop. of ~Δs) 15. [Image of a triangle with various line segments] ΔHIF ~ ΔJIG (AA ~); ΔJEH ~ ΔJEF (AA ~); ΔJIH ~ ΔJEH (Trans. Prop. of ~Δs) 16. [Image of a triangle with various line segments] ΔTUV ~ ΔVWU (AA ~); ΔTUV ~ ΔVXY (AA ~); ΔVWU ~ ΔVXY (Trans. Prop. of ~Δs); ΔVWZ ~ ΔVWU (AA ~); ΔVWZ ~ ΔVXY (Trans. Prop. of ~Δs); ΔTUV ~ ΔVWZ (Trans. Prop. of ~Δs); ΔTUV ~ ΔVXY (Trans. Prop. of ~Δs); ΔVWZ ~ ΔVXY (Trans. Prop. of ~Δs); ΔTUV ~ ΔVWZ (Trans. Prop. of ~Δs); ΔVWZ ~ ΔVXY (Trans. Prop. of ~Δs); ΔTUV ~ ΔVXY (Trans. Prop. of ~Δs) 17. [Image of a triangle with various line segments] ΔJKL ~ ΔJIH (AA ~); ΔJIK ~ ΔJIH (AA ~); ΔJKL ~ ΔJIK (Trans. Prop. of ~Δs); ΔJKL ~ ΔJHI (Trans. Prop. of ~Δs); ΔJIK ~ ΔJHI (Trans. Prop. of ~Δs)