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?

<p>A triangle with one right angle</p> Signup and view all the answers

What is an equilateral triangle?

<p>A triangle with 3 congruent sides</p> Signup and view all the answers

What defines an isosceles triangle?

<p>A triangle with at least two congruent sides</p> Signup and view all the answers

What is the Triangle Angle-Sum Theorem?

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

Define the remote interior angles of a triangle.

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

What is a flow proof?

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

What is a corollary?

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

Congruent polygons have the same shape and size.

<p>True (A)</p> Signup and view all the answers

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

<p>If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent. (C)</p> Signup and view all the answers

What is included angle?

<p>The angle formed by two adjacent sides of a polygon. (A)</p> Signup and view all the answers

Define transformation in geometry.

<p>An operation that maps an original geometric figure onto a new figure. (A)</p> Signup and view all the answers

What is the distance formula?

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

What defines complementary angles?

<p>Two angles whose sum is 90 degrees.</p> Signup and view all the answers

What defines supplementary angles?

<p>Two angles whose sum is 180 degrees.</p> Signup and view all the answers

What is an acute angle?

<p>An angle that measures less than 90 degrees.</p> Signup and view all the answers

What is a right angle?

<p>An angle that measures 90 degrees.</p> Signup and view all the answers

Define an obtuse angle.

<p>An angle that measures between 90 and 180 degrees.</p> Signup and view all the answers

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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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