Chapter 4 Notes

4.1 – Triangles and Angles

A Triangle  Three segments joining three noncollinear points. Each point is a VERTEX of the triangle. Segments are SIDES! A B C Scalene – No congruent sides Isosceles – At LEAST 2 congruent sides Equilateral – All sides congruent

Acute – 3 acute angles Obtuse – One obtuse angle Right – One right angle Equiangular – all angles congruent

A B C

1 3 TRIANGLE SUM THEOREM The sum of the measures of the angles of a triangle is

Exterior Angles Theorem The measure of an exterior angle of a triangle equals the sum of the two remote interior angles. (remote means nonadjacent) Statement Reason

Corollary to triangle sum theorem: Acute angles of a right triangle are complementary. All angles 180, if one is 90, the other two add up to 90, and are complementary

4.2 – Congruence and Triangles

When TWO POLYGONS have the same size and shape, they are called CONGRUENT! Their vertices and sides must all match up to be congruent. When two figures are congruent, their corresponding sides and corresponding angles are congruent. Identical twins!

A C B D E F Name all the corresponding parts and sides, then make a congruence statement. If you notice, the way you name the triangle is important, all the CORRESPONDING SIDES must line up!

3 rd Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the 3 rd angles are congruent. A B C D E F

M A H T G E YO Find xFind y

Note, triangles also have the following properties of congruent: Reflexive, symmetric, and transitive.

4.3 – 4.4 Proving Triangles are Congruent

A C B D E F SSS Congruence Postulate – If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

A C B D E F 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. Included means IN BETWEEN

A C B D E F ASA Congruence Postulate – If two angles and the included side of one triangles are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

A C B D E F AAS Congruence Theorem – If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two trianges are congruent.

A C B D E F

A C B E F

Helpful things for the future! Reflexive sides Reflexive angles A B D C F E H G When you see shapes sharing a side, you state that fact using the reflexive property of congruence!

A C B D

Draw and write down if the triangles are congruent, and by what thrm\post

Proofs! The way I like to think about it to look at all the angles and sides, and don’t be fooled by the picture. A B C D E Tips, label the diagram as you go along.

A B D E

F E H G B C What about the angle?

Use SSS Congruence Postulate to show that A C B D E F (-3, -2) (-4, -3) (-5, 1) (2, 2) (5, 4) (1, 3)

A B D E Tips, label the diagram as you go along.

U D C K S

C B A D E

C B A D E

4.5 – Using Congruent Triangles

A B C D E

Some Ideas that may help you. If they want you to prove something, and you see triangles in the picture, proving triangles to be congruent may be helpful. If they want parallel lines, look to use parallel line theorems (CAP, AIAT, AEAT, CIAT) Know definitions (Definition of midpoint, definition of angle bisectors, etc.) Sometimes you prove one pair of triangles are congruent, and then use that info to prove another pair of triangles are congruent.

A B D E

U D C K S

C B A D E

G N A L E

You try this classic proof! A B C D E

4.6 – Isosceles, Equilateral, and Right Triangles.

Bring book Tuesday We will go over what’s going to be on Wednesday’s Quiz at end of Tuesday lesson

Vertex Angle Base Angles BASE LEGS Remember, definition of isosceles triangles is that AT LEAST two congruent sides.

Base angles theorem – If two sides of a triangle are congruent, then the base angles are congruent. Converse of Base angles theorem – If base angles are congruent, then the two opposite sides are congruent.

Corollary 1 – An equilateral triangle is also equiangular (Use isosceles triangle theorem multiple times with transitive) Corollary 2 – An equilateral triangle has three 60 degree angles (Use corollary 1 and angle of triangle equals 180)

Hypotenuse Leg Theorem (HL) – If the hypotenuse and ONE of the legs of a RIGHT triangle are congruent, then the triangles are congruent.

A C B D

Draw and write down if the triangles are congruent, and by what thrm\post

Def of isosceles triangle

C B A D

4.7 – Triangles and Coordinate Proof

Given a right triangle with one vertex (-20, -10), and legs of 30 and 40, find two other vertices, then find the length of the hypotenuse.

Given a vertex of a rectangle at the origin, find three other possible vertices if the base is 15 and the height is 10 for a rectangle. Then find the area.

Given the coordinates, prove that the AC is the angle bisector of BCD B CD A

(a, b) (__,__) (h,k) (__,__) (j,__) (d, k) (__,__) (__, k) (a,__)

Picking convenient variable coordinates, prove that the diagonals of a rectangle are congruent.