GEOMETRY – CHAPTER 4Congruent Triangles

4.1 – Apply Angle Sum Properties• Triangle

• Polygon with three sides & three vertices

•Triangles can be classified by side and angles

Classifying Triangles by Sides

Scalene Iscoceles Equilateral

No congruent sides

Two congruent sides

All sides congruent

Classifying Triangles by Angles

Acute Right Obtuse Equiangular

3 acute angles (< 90)

1 right angle (= 90)

1 obtuse angle (> 90)

3 congruent angles

Example 2• Classify ∆PQO by its sides, then determine if the triangle

is right.

• Points are:• P (-1, 2)• Q (6, 3)• O (0, 0)

GP: #1-2

Angles• Interior Angles

• Angles on the inside of the triangle (there are three)

• Exterior Angles• Angles that form linear pairs with interior angles (there are 6)

Theorems• 4.1 – Triangle Sum Theorem

• The sum of the measure of the interior angles of a triangle is 180°

• 4.2 – Exterior Angle Theorem• The measure of an exterior angle of a triangle is equal to the sum

of the measures of the two nonadjacent interior angles

Example #3 p. 209

Corollary to a theorem• Corollary to a theorem

• Statement that can be proved easily using the theorem

• Corollary to the Triangle Sum Theorem• The acute angles of a right triangle are complementary

• Example 4• A tiled staircase forms a right triangle. The measure of one acute

angle in the triangle is twice the measure of the other. Find the measure of each acute angle

GP #3 & 5 p. 210

4.2 – Apply congruence & triangles• Two geometric figures are congruent if they have exactly

the same size and shape

• Congruent figures• All parts of one figure are congruent to the corresponding parts of

the other figure (corresponding sides & corresponding angles)

• Congruence Statements• Be sure to name figures by their corresponding vertices!

examples• Example 1

• Writing a congruence statement and identifying all congruent parts

• Example 2• Using properties of congruent figures

GP #1-3 p. 216

Third angles theorem• Theorem 4.3 – Third Angles Theorem

• If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent

Using third angles theorem• Example 4

• Find m< BDC

A

C

B

D

N45°

30°

GP #4-5 p. 217

Properties of congruent triangles• The properties of congruence that are true for segments

and angles are also true for triangles

• Theorem 4.4 – Properties of Congruent Triangles• Reflexive property

• Symmetric property

• Transitive property

4.3 – relate transformations & congruence

• Rigid motion• Transformation that preserves length, angle measure, and area• Examples of rigid motions (isometry): translations, reflections,

rotations

• Congruent figures and Transformations• Two figures are congruent if and only if one or more rigid motions

can be used to move one figure onto the other. If any combination of translations, reflections, and rotations can be used to move one shape onto the other, the figures are congruent

4.4 – Prove triangles congruent by SSS

• Postulate 19 – Side-Side-Side (SSS) Congruence Postulate• If three sides of one triangle are congruent to three sides of a

second triangle, then the two triangles are congruent

Example 1• Use the SSS congruence postulate

• GP #1-3 p. 232

4.5 – congruence by SAS and HL• Postulate 20 – Side-Angle-Side (SAS) Congruence

Postulate• 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

Right triangles• In a right triangle, the sides adjacent to the right angles

are called the legs

• The side opposite the right angle is called the hypotenuse

• Theorem 4.5 – Hypotenuse-Leg (HL) Congruence Theorem• If the hypotenuse and a leg of a right triangle are congruent to the

hypotenuse and a leg of a second right triangle, then the two triangles are congruent

4.6 – Prove using ASA & AAS• Postulate 21 – Angle-Side-Angle (ASA) Congruence

Postulate• If two angles and the included side of one triangle are congruent to

two angles and the included side of a second triangle, then the two triangles are congruent

AAS theorem• Theorem 4.6 – Angle-Angle-Side (AAS) Congruence

Theorem• If two angles and a non-included side of one triangle are congruent

to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent

Triangles postulates &theorems• 5 methods for proving that triangles are congruent

SSS SASHL (right

angles only)ASA AAS

All 3 sides are congruent

Two sides and the included angle are congruent

Hypotenuse and one of the legs are congruent

Two angles and the included side are congruent

Two angles and a non-included side are congruent

4.7 – use congruent triangles• Congruent triangles have congruent corresponding parts

• If two triangles are congruent, their corresponding parts must be congruent as well

Euclid’s river example

4.8 – use isosceles and equilateral triangles

• Legs• Two congruent sides of an isosceles triangle

• Vertex angle• Angle formed by the legs

• Base• Third side of an isosceles triangle

• Base angles• Angles adjacent to the base (opposite the legs)

Isosceles triangles theorem• Theorem 4.7 – Base Angles Theorem

• If two sides of a triangle are congruent, then the angles opposite them are congruent

• Theorem 4.8 – Converse of Base Angles Theorem• If two angles of a triangle are congruent, then the sides opposite

them are congruent

Example 1• Name two congruent angles

F

E

D

GP #1-2 p. 264

Corollaries• Corollary to the Base Angles Theorem

• If a triangle is equilateral, then it is equiangular

• Corollary to the Converse of Base Angles Theorem• If a triangle is equiangular, then it is equilateral

• Example 2• If a triangle is equilateral, what is the measure of each angle?

• Example 3 – on board GP #5 p. 266