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Chase Beard 4-6 Congruence in Right Triangles In a right triangle, a triangle with one right angle, there is a unique way to prove triangles congruent. This way is called the Hypotenuse-Leg Theorem or HL Theorem. Parts of Right Triangles In right triangles, the side opposite the right angle, an angle that’s measure is 90°, is the Hypotenuse, and the other two sides including the right angle are Legs. Hypotenuse Leg Leg Theorem 4-6: HL Theorem If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Proof: What you will need to know: CPCTC: Corresponding Parts of Congruent Triangles are Congruent; Transitive Property of Congruence: If line AB is congruent to line CD, and line CD is congruent to line EF, then line AB is Congruent to line EF; Def. of a right triangle: a triangle with one right angle; Def. of right angle: an angle who’s measure is 90°; Theorem 4-3: Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent; SAS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent; AAS Theorem: If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of another triangle, then the triangles are congruent. Given: ΔPQR and ΔXYZ are right triangles, with right angles Q and Y respectively. Lines PR and XZ are congruent, and lines PQ and XY are congruent. Prove: ΔPQR is congruent to ΔXYZ X P Figure 1 Q R Y Z Proof: On ΔXYZ at the right, draw ray ZY. Mark point S as shown in figure 2 so that YS = QR. Then, ΔPQR is congruent to ΔXYS by SAS. By CPCTC, line PR is congruent to line XS. It is given that line PR is congruent to line XZ, so line XS is congruent to line XZ by the Transitive Property of Congruence. By the Isosceles Triangle Theorem, <S is congruent to <Z, so ΔXYS is congruent to ΔXYZ by AAS. Therefore, ΔPQR is congruent to ΔXYZ by the Transitive Property of Congruence. X Figure 2 S Y Z Examples: Example 1 What you will need to know: Def. of perpendicular lines: two lines that intersect to form right angles; Def. of a right triangle: a triangle with one right angle; Def. of right angle: an angle whose measure is 90°; Reflexive Property of Congruence: Line AB is congruent to line AB. Given: line PS is perpendicular to line SQ; line RQ is perpendicular to line QS; line PQ is P congruent to line RS Prove: ΔPSQ is congruent to ΔRQS Proof: Statements Reasons Line PS is perpendicular to line SQ; Line RQ is perpendicular to line QS; line PQ is congruent to line RS <RQS and <QSP are right angles. ΔPSQ and ΔRQS are right triangles ΔPSQ is congruent to ΔRQS S Given R Def. of Perpendicular Def. of a right triangle Line QS is congruent to line QS Q Reflexive Property of Congruence HL Theorem Example 2 What you will need to know: Def. of perpendicular lines: two lines that intersect to form right angles; Def. of a right triangle: a triangle with one right angle; Def. of right angle: an angle whose measure is 90°; Reflexive Property of Congruence: Line AB is congruent to line AB. Given: line LN is perpendicular to line KM; line KL is congruent to line ML Prove: ΔKLN is congruent to ΔMLN L K Proof: Line LN is perpendicular to line KM Given N M <LNM and <LNK are right angles Def. of perpendicular ΔKLN and ΔMLN are right triangles Def. of right triangle Line KL is congruent to line ML Given Line LN is congruent to line LN Reflexive Property of Congruence ΔKLN is congruent to ΔMLN HL Theorem