Chapter 4: Triangles and Congruence

4.1: Classifying triangles

Vocabulary

An equilateral triangle is any triangle with three equal sides and angles.

An isosceles triangle is any triangle with two equal sides and two equal angles.

A scalene triangle is any triangle with three unique sides and three unique angles.

A right triangle is any triangle with a right angle.

An acute triangle has all angles less than 90 degrees.

An obtuse triangle has one angle greater than 90 degrees.

Note

Equal sides means equal angles.

4.2: Angles of triangles

Vocabulary

The triangle angle-sum theorem states that all angles in a triangle add to 180 degrees.

An exterior angle of an triangle is the angle created between two lines that make up a triangle.

A remote angle is an angle not adjacent to the exterior angle.

The exterior angle theorem sates that the exterior angle adds up to the sum of the interior angles.

Examples

The triangle angle-sum theorem states that $\mathrm{\angle }1$, $\mathrm{\angle }2$, and $\mathrm{\angle }3$ all add up to 180 degrees. $\mathrm{\angle }1$ and $\mathrm{\angle }3$ are the remote angles of $\mathrm{\angle }4$, an exterior angle. The exterior angle theorem states the $m\mathrm{\angle }1+m\mathrm{\angle }3=m\mathrm{\angle }4$.

4.3: Congruent triangles

Vocabulary

Congruent figures have the same size and shape.

Writing Congruent Statements

4.4: Proving triangles congruent SSS SAS

Vocabulary

SSS (Side Side Side) states that if all sides of a triangle are equal, then the triangles are congruent.

SAS (Side Angle Side) states that that if two sides of a triangle and the interior angle by those sides are equal to another triangle’s side, side and angle, then the triangles are congruent.

Warning

There is no AAA (Angle Angle Angle) or ASS (Angle Side Side).

Example

$\mathrm{△}ABC\cong \mathrm{△}DEF$ since SAS ($\overline{AB}\cong \overline{DC};\overline{CB}\cong \overline{EC};\mathrm{\angle }B\cong \mathrm{\angle }C$) or SSS ($\overline{AB}\cong \overline{DC};\overline{CB}\cong \overline{EC};\overline{CA}\cong \overline{ED}$).

4.5: Proving triangles congruent ASA AAS

Vocabulary

ASA (Angle Side Angle) states that if two triangles share a congruent side that touch two congruent angles, then the two triangles are congruent.

AAS (Angle Angle Side) states that if two triangles share a congruent side that both touch one congruent angle and the triangles also has an angle that does not touch the side, then the two triangles are congruent.

4.6: Isosceles and Equilateral Triangles

Vocabulary

An isosceles triangle has exactly two congruent sides with two congruent angles.

An equilateral triangle has exactly three congruent sides and angles.

CPCTC states that all sides and angles in two congruent triangles are congruent.

Congruent Parts

If $\mathrm{△}ABC$ was an equilateral triangle, then all sides would be congruent. If $\mathrm{△}DEF$ was an isosceles, then $\overline{DF}\cong \overline{DE};\mathrm{\angle }DFE\cong \mathrm{\angle }DEF$.

4.7: Congruence transformations

:)

4.8: Triangles and Coordinate Proof

Horary, no vocabulary!

How to identify triangles based on coordinates

1. Find the distance between all points using the distance formula, more information in 1.2: Line Segments and Distance.

2. Check for equal sides

3. Based on the number of equal sides find what kind of a triangle it is, scalene (no sides are equal), isosceles (two equal sides), or equilateral (three equal sides).

Before Final Notes :)

The isosceles triangle theorem states that, if two sides of a triangle are congruent, then the angles opposite those sides are congruent. The congruent angles are called the base angles.