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