Congruent Triangles, Chapter 4

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Questions and Answers

What is an acute triangle?

A triangle with one obtuse angle

A triangle with one right angle

A triangle with 3 sides

A triangle with 3 acute angles (correct)

What defines an equiangular triangle?

A triangle with 3 congruent angles

What is the definition of an obtuse triangle?

A triangle with one obtuse angle

What defines a right triangle?

A triangle with one right angle Signup and view all the answers

What is an equilateral triangle?

A triangle with 3 congruent sides Signup and view all the answers

What defines an isosceles triangle?

A triangle with at least two congruent sides Signup and view all the answers

What is the Triangle Angle-Sum Theorem?

The sum of the measures of the angles of a triangle is 180. Signup and view all the answers

Define the remote interior angles of a triangle.

The two angles that are not adjacent to a given exterior angle of a triangle. Signup and view all the answers

What is a flow proof?

A type of proof that uses statements written in boxes and arrows to show the logical progression of an argument. Signup and view all the answers

What is a corollary?

A theorem with a proof that follows as a direct result of another theorem. Signup and view all the answers

Congruent polygons have the same shape and size.

True Signup and view all the answers

What does the Side-Side-Side (SSS) Congruence theorem state?

If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent. Signup and view all the answers

What is included angle?

The angle formed by two adjacent sides of a polygon. Signup and view all the answers

Define transformation in geometry.

An operation that maps an original geometric figure onto a new figure. Signup and view all the answers

What is the distance formula?

d = √[(x₂ - x₁)² + (y₂ - y₁)²] Signup and view all the answers

What defines complementary angles?

Two angles whose sum is 90 degrees. Signup and view all the answers

What defines supplementary angles?

Two angles whose sum is 180 degrees. Signup and view all the answers

What is an acute angle?

An angle that measures less than 90 degrees. Signup and view all the answers

What is a right angle?

An angle that measures 90 degrees. Signup and view all the answers

Define an obtuse angle.

An angle that measures between 90 and 180 degrees. Signup and view all the answers

Study Notes

Types of Triangles

Acute Triangle: Contains three angles, each less than 90 degrees.
Obtuse Triangle: Has one angle greater than 90 degrees.
Right Triangle: Features one angle exactly at 90 degrees.
Equilateral Triangle: All three sides are equal in length; also has three congruent angles (60 degrees each).
Isosceles Triangle: At least two sides are equal, leading to two congruent angles.
Scalene Triangle: No sides are congruent, and all angles are different.

Triangle Congruence

Congruent Polygons: Two shapes are congruent if their corresponding sides and angles match in measure.
Side-Side-Side (SSS) Congruence: Triangles are congruent if all three sides of one are equal to the three sides of another.
Side-Angle-Side (SAS) Congruence: Congruency is established when two sides and the included angle of one triangle match those of another.
Angle-Side-Angle (ASA) Congruence: Two triangles are congruent if two angles and the included side are congruent.
Angle-Angle-Side (AAS) Congruence: If two angles and the non-included side of one triangle match those of another, the triangles are congruent.
Leg-Leg (LL) Congruence: Specifically for right triangles, both legs must be congruent.
Hypotenuse-Angle (HA) Congruence: The hypotenuse and one acute angle of a right triangle must be congruent to those of another.
Leg-Angle (LA) Congruence: One leg and an acute angle of a right triangle must match those of another.
Hypotenuse-Leg (HL) Congruence: In right triangles, the hypotenuse and one leg must be congruent between the two triangles.

Angle Relationships

Triangle Angle-Sum Theorem: The total measure of angles in a triangle is always 180 degrees.
Exterior Angle: Formed by extending one side of the triangle; it is supplementary to the adjacent interior angle.
Remote Interior Angles: Two interior angles that are not adjacent to a given exterior angle of a triangle.

Isosceles Triangle Theorem

Isosceles Triangle Theorem: States that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent.
Converse of the Isosceles Triangle Theorem: If two angles in a triangle are congruent, their opposite sides are also congruent.

Proof Techniques

Flow Proof: A structured proof using boxes and arrows to illustrate logical connections.
Two-Column Proof: Presents statements and the corresponding reasons in a two-column format for clarity.
Coordinate Proof: Involves plotting figures on a coordinate plane for proof.

Transformations

Transformation: An operation that maps a figure onto another.
Rigid Transformation: Any transformation that preserves the shape and size of the figure.
Isometry: Another term for rigid transformations.
Reflections: A flip of the figure over a line.
Translations: A slide in a specific direction.
Rotations: A turn around a fixed point.

Angle Relationships

Complementary Angles: Two angles sum up to 90 degrees.
Supplementary Angles: Two angles sum up to 180 degrees.

Distance and Midpoint Formulas

Distance Formula: Calculated as (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).
Midpoint Formula: Determines the midpoint between two coordinates as (\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)).

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Description

Test your knowledge of congruent triangles with this quiz based on Chapter 4 of McGraw-Hill materials. You'll encounter definitions for various types of triangles such as acute, obtuse, and right triangles, as well as equiangular triangles. Perfect for students preparing for geometry assessments.