If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Side-Angle-Side (SAS) congruence postulate: Right triangles – one angle measures exactly 90°.Equilateral triangles – all angles measure 60°.Obtuse triangles – one angle measures between 90° and 180°.Acute triangles – all angles measures less than 90°.Solve the problems on HL congruence of triangles.Solve the problems on SAS congruence of triangles.Understand and apply the HL congruence theorem.Identify the properties of right triangles.Understand and apply the SAS congruence postulate.In a Δ ABC, if AB = AC and ∠B = 70°, find ∠A.You get out your mathematical detective's magnifying glass and notice that ∠O and ∠G are marked with the tell-tale little squares, □, indicating right angles.Īha! These are two right triangles, because by definition a right triangle has one right angle. You have two suspicious-looking triangles, △MOP and △RAG. So, we have proven the HL Theorem, and can use it confidently now! HL theorem practice proof

We originally used the isosceles triangle to find the hypotenuse and a single leg congruent, and from that, we built proof that both triangles are congruent. So, all three interior angles of each right triangle are congruent, and all sides are congruent. ∠AJC ≅ ∠CJK ( side JC was the angle bisector of original ∠AJK) ∠A ≅ ∠K (they were angles opposite to the legs in accordance with the Isosceles Triangle Theorem) Now verify that AC ≅ CK and all the interior angles are congruent:ĪC ≅ CK (the altitude of the base of an isosceles triangle bisects the base, since it is by definition the perpendicular bisector) So, we have one leg and a hypotenuse of △JAC congruent to the corresponding leg and hypotenuse of △JCK. We know by the reflexive property that side JC ≅ JC (it is used in both triangles), and we know that the two hypotenuses, which began our proof as equal-length legs of an isosceles triangle, are congruent. We have two right triangles, △JAC and △JCK, sharing side JC. We have two right angles at Point C, ∠JCA and ∠JCK. That altitude, JC, complies with the Isosceles Triangle Theorem, which makes the perpendicular bisector of the base the angle bisector of the vertex angle. Recall that the altitude of a triangle is a line perpendicular to the base, passing through the opposite angle. Can you guess how?Ĭonstruct an altitude from side AK. We are about to turn those legs into hypotenuses of two right triangles. We know by definition that JA ≅ JK, because they are legs. To prove that two right triangles are congruent if their corresponding hypotenuses and one leg are congruent, we start with an isosceles triangle. Once proven, it can be used as much as you need. We have to enlist the aid of a different type of triangle. So we have to be very mathematically clever.

Of course you can't, because the hypotenuse of a right triangle is always (always!) opposite the right angle. Hold on, you say, that so-called theorem only spoke about two legs, and didn't even mention an angle?Īha, have you forgotten about our given right angle? Every right triangle has one, and if we can somehow manage to squeeze that right angle between the hypotenuse and another leg. The Hypotenuse Leg Theorem, or HL Theorem, states If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. The included angle has to be sandwiched between the sides. Recall the SAS Postulate used to prove congruence of two triangles if you know congruent sides, an included congruent angle, and another congruent pair of sides. The converse of this, of course, is that if every corresponding part of two triangles are congruent, then the triangles are congruent. HL Theorem CPCTCĬPCTC reminds us that, if two triangles are congruent, then every corresponding part of one triangle is congruent to the other. Usually you need only three (or sometimes just two!) parts to be congruent to prove that the triangles are congruent, which saves you a lot of time. Notice the squares in the right angles.Įvery part of one triangle is congruent to every matching, or corresponding, part of the other triangle. Notice the hash marks for the three sides of each triangle. Notice the hash marks for the two acute interior angles. Here are two congruent, right triangles, △PAT and △JOG. It is shortened to CPCTC, which is easy to recall because you use three Cs to write it. CPCTCĬPCTC is an acronym for corresponding parts of congruent triangles are congruent. The longest side of a right triangle is called its hypotenuse.

Sas theorem