lesson twelve: congruence the right way. congruence as we have discovered, there are many congruence...

LESSON TWELVE:

CONGRUENCE THE RIGHT WAY

CONGRUENCE

• As we have discovered, there are many congruence theorems for all types of triangles.

• As we will find out in the coming days, some types or triangles have unique properties.

RIGHT TRIANGLE CONGRUENCE• Right Triangles are one type that have special

properties.• We will discuss four of them.• As we introduced these, notice the similarities

between our original congruence postulates.

RIGHT TRIANGLE CONGRUENCE

• The first theorem for right angles we’ll learn is the Leg-Leg Congruence Theorem.

• This says that if the legs of one right triangle, are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

RIGHT TRIANGLE CONGRUENCE

RIGHT TRIANGLE CONGRUENCE• The next theorem for right angles is the

Hypotenuse-Angle Congruence Theorem.• This says if the hypotenuse and acute angle of

a right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle then the two triangles are congruent.

RIGHT TRIANGLE CONGRUENCE

RIGHT TRIANGLE CONGRUENCE

• The third theorem in our lineup is the Leg-Angle Congruence Theorem.

• This says that if one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent.

RIGHT TRIANGLE CONGRUENCE

RIGHT TRIANGLE CONGRUENCE

• Finally, the Hypotenuse-Leg Theorem.• This says that if the hypotenuse and a leg of

one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

RIGHT TRIANGLE CONGRUENCE

RIGHT TRIANGLE CONGRUENCE

• We can use these theorems in proofs the same way as we can use any of the other properties, postulates or theorems we have learned.

RIGHT TRIANGLE CONGRUENCE

• For example…• Given that MC is the perpendicular bisector to

AB, prove that ∆AMC ∆BMC.

A M

C

B

STATEMENTS

1. MC is a perpendicular bisector of AB

2. AMMB3. mAMC = 90⁰, mBMC = 90

⁰4. ∆AMC and ∆BMC are Right

Triangles5. CM CM6. ∆AMC ∆BMC

JUSTIFICATION

1. Given

2. Definition of Bisector3. Definition of Perpendicular

Lines.4. Definition of Right

Triangles5. Reflexive Property 6. LL Congruence

A M

C

B

RIGHT TRIANGLE CONGRUENCE• The only difference between these proofs and

the ones we have already done is that we need to occasionally prove that we are, in fact, dealing with a right triangle.

• Sometimes it is given, sometimes not.• Without that, we cannot use any of these

theorems.