Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors

Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors.

Review. TRIANGLE CONGRUENCE POSTULATES AND THEOREMS FOR ANY TRIANGLES FOR RIGHT TRIANGLES SAS Congruence Postulate ASA Congruence Postulate SSS Congruence Postulate AAS Congruence Theorem HA Congruence Theorem LL Congruence Theorem LA Congruence Theorem HL Congruence Postulate.

State the postulate or theorem to prove that the triangles are congruent..

. TRIANGLE CONGRUENCE POSTULATES c by SSS Congruence Postulate 1 by SAS Congruence Postulate c by ASA Congruence Postulate s by AAS Congruence Theorem 1.

. RIGHT TRIANGLE CONGRUENCE THEOREMS HA Congruence Theorem AACB AOST LA Congruence Theorem b AACB AOST AC=OS L LL Congruence Theorem AACB AOST ÄC=ÖS L HL Congruence Postulate AACB AOST ÄÉæöiH CB=SÉ L.

. Definition of terms. ANGLE PISECTOR Angle bisector is a line segment or ray that divides an angle into two equal parts. Given: Any LABC To construct: bisector of LABC AFTER BEFORE c c.

PERPENDICULAR LINES Perpendicularity is not limited to lines only. Segments and rays can also be perpendicular. Perpendicular Bisector of a segment is a line (or a ray or another segment) perpendicular to the segment at its midpoint. c CD is the perpendicular bisector of AB 1. Since CD is a perpendicular bisector, it will form right angles which are LCDA and LCDB. 2. CD bisects AB, therefore Äb will be divided into two congruent segments which are AD and Db. Conclusion: LCDA and LCDB are right angles. ÄD DB What conclusion can you make?.

. Constructing Angle Bisector using Ovo congruent right triangles 2 (mz-2) 1. 2. 4. Given tvvo congruent triangles by ASA Congruence Postulate, other corresponding parts that are congruent- detern•ine o 3 2 1 6 5 4 Using Corresponding Parts Of Congruent Triangles are Congruent (CPCTC), the the followring corresponding parts are congruent. MO MP NO NP Put the tvvo triangles together in such a vvay that a pair Of corresponding sides coincide. Change the label Of the overlapping sides. See the thicker line. Deter-tmine the follosving: 1 4 3 5 2 6 cornn•on side or the side shared by the triangles- The conmn•on side is MN . adjacent angles forrned and its relationship The adjacent angles are zl and z4; z2 and congruent as they are corresponding parts Of the Deter-rnine the relationship Of any one Of adjacent each other. z6. These pair Of angles are congruent triangles. angles to the Of their rneasures. mzOMP = + rnz-4 mzOMP = 2 (mzl) mzorv1P — 1/2 (mzorv1P) mzONP mzONP = Inz-2 + In z 2 1/2 (mz-0NP) Thus, any one Of the adjacent angles is half Of the angle. Since the pairs Of adjacent angles are congruent and each angle is either half Of ZONA P or half of ZOBJP, then it MN divides both LOMP and LON P congruently. Thus, the side of MN bisects both zorv1P and ZONP. Hence, side is an angle bisector. Angle bisector is a line, ray, or segn•ent that divides the angle into congruent angles..

Constructing Perpendicular Lines using two congruent right triangles Example 1: 1. 2. 3. Given two congruent right triangles by L-AA Congruence Theorem, determine the other corresponding parts that are congruent. x x c Using Corresponding Parts of Congruent Triangles are Congruent (CPCTC), the following corresponding parts are congruent. LXAY LDXC ÄX=XD ÄY=XC Put the two triangles side by side in such a way that the vertices labeled with X coincide. Determine the relationship of LAXY and LCXD. nuA + mLAXY + nuAYX = 1 800 mLA + mLAXY + 900 1 800 mzA + = 1 800 - 900 mLA + mLAXY = 900 nuAXY = 900 - mzA c x Since LA LCXD, then mLAXY = 900 - mLCXD or mLAXY + mLCXD = 900 Thus, LAXY and LCXD are complementary. Since LAXY and LCXD intersect at a common point X and the sum of their measures is equal to 90 degrees, then ÄX is perpendicular to or Xb is perpendicular to ÄX by the definition of perpendicularity. Perpendicular lines are lines that intersect at a common point forming 900 angle..

Example 2: Steps: l. 2. 3. Given two congruent triangles by L-L Congruence Theorem, determine the other corresponding parts that are congruent. Using Corresponding Parts of Congruent Triangles are Congruent (CPCTC), the following corresponding parts are congruent. LABC LDBC, LACB LDCB, Put the two triangles side by side in such a way that a pair of corresponding sides coincide. Determine the common side shared by two triangles and the pairs of adjacent angles. Common Side: BC Adjacent Angles: Pair 1: LABC and LDBC Pair 2: LACB and LDCB B B cc c.

4. Determining relationships a. b. c. Determine the relationship of the adjacent angles. Each pair of adjacent angles in no. 3 are congruent. Pair 2 are both right angles. Determine the relationship of the common side and the vertex angle. The common side BC is the angle bisector of the vertex angle since LABC LDBC. Determine the relationship of the common side and the base of the larger triangle. They are perpendicular lines since the angles formed, LACB and LDCB are right angles. The common side also divides the base of the larger triangle at C into two congruent segments. Since the common side BC intersects AD perpendicularly at C and divides it into two congruent parts resulting to AC DC. Hence, side BC is a perpendicular bisector ofÄD. Perpendicular bisector is a line that bisects another line segment at a right angle through the intersection point..

PRACTICE TASK.

Practice Task No. 1 Use AABK LACK to answer the questions that follow. l. What triangle congruence postulate is illustrated by the two triangles? 2. What are the corresponding congruent parts? 3. Ifthe two triangles are placed together in such a way that side AK of AABK and side AK of AACK coincided with each c other, what new figure is formed? 4. 5. 6. 7. 8. Do the sides of the two triangles that coincide appear to be congruent? Why? What is the common side shared by the two triangles? What are the pair of adjacent angles? Are each pair of adjacent angles congruent? Justify. What does AK do to LBAC and LBKC?.

CHECK YOUR ANSWER.

Practice Task No. 1 1. 2. 3. 4. 5. 6. 7. 8. SAS Congruence Postulate ÄR ÄR, LAKB LAKC, CR a kite / kite BACK Yes. It is the common side. LKAB LKAC, LAKB LAKC Yes. They are pairs of corresponding parts of congruent triangles. AK bisects both LBAC and LBKC, hence AK is an angle bisector..

EVALUATION.

lv. EVALUATION Choose the letter of the best answer. 1. Which of the following is TRUE about an angle bisector? a. It divides an angle into halves. b. It divides an angle at 450. For items 2 and 3, use AACT at the right: Given: AO is the perpendicular bisector of CT 2. Which of the following angles are congruent? c. It divides an angle at 900. d. It divides an angle into three parts. c a. LCOA and LTOA b. LOAT and LOCA c. LOAT and LATO d. LAOC and LTAO 3. What triangle congruence postulate/theorem justifies AAOC AAOT? b. ASA d. SSS a. AAS For items 4 and 5, use the figure at the right to answer the questions that follow: 4. What is the common side of ASIT and ASET? 5. Which of the following best describes the adjacent angles in the figure? The adjacent angles are congruent and are s a. both right angles b. supplementary angles c. complementary angles d. corresponding angles of the congruent triangles.