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Questions and Answers

What is a triangle?

A figure formed by three segments connecting three noncollinear points.

What are the parts of a triangle?

Three sides, three vertices, and three angles.

How is a triangle named?

A triangle is named by using the three letters that name its vertices.

What is an acute triangle?

<p>An acute triangle is a triangle in which every angle is acute.</p> Signup and view all the answers

What is a right triangle?

<p>A right triangle is a triangle that contains exactly one right angle.</p> Signup and view all the answers

What is an obtuse triangle?

<p>An obtuse triangle is a triangle that contains exactly one obtuse angle.</p> Signup and view all the answers

What is an equilateral triangle?

<p>An equilateral triangle is a triangle with three congruent sides.</p> Signup and view all the answers

What is an isosceles triangle?

<p>An isosceles triangle is a triangle with at least two congruent sides.</p> Signup and view all the answers

Triangles that are equilateral are also isosceles.

<p>True</p> Signup and view all the answers

What is a scalene triangle?

<p>A scalene triangle is a triangle with no two congruent sides.</p> Signup and view all the answers

What is the rule for what side lengths can form a triangle?

<p>The sum of the lengths of any two sides must be greater than the length of the third side.</p> Signup and view all the answers

What is the sum of all the angles in a triangle?

<p>180º</p> Signup and view all the answers

If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are ____.

<p>congruent</p> Signup and view all the answers

How many exterior angles does a triangle have?

<p>6</p> Signup and view all the answers

What are remote interior angles?

<p>Angles that do not touch the exterior angle in question.</p> Signup and view all the answers

What is the relationship between an exterior angle and its two remote interior angles?

<p>The measures of the sum of the two remote interior angles equal the measure of the exterior angle.</p> Signup and view all the answers

Congruent triangles have exactly the same ____ and ____.

<p>size; shape</p> Signup and view all the answers

The three congruence transformations are _______, _______, and _______.

<p>translating; rotating; reflecting</p> Signup and view all the answers

How can you find the corresponding parts of two congruent triangles?

<p>By aligning them perfectly on top of each other.</p> Signup and view all the answers

If two triangles are congruent, all three pairs of corresponding _____ and all three pairs of corresponding _____ are also congruent.

<p>sides; angles</p> Signup and view all the answers

What are congruent triangles?

<p>Congruent triangles are triangles whose corresponding parts are congruent.</p> Signup and view all the answers

What is CPCTC?

<p>Corresponding Parts of Congruent Triangles are Congruent.</p> Signup and view all the answers

What is the reflexive property of congruence?

<p>Every triangle is congruent to itself.</p> Signup and view all the answers

What is the symmetric property of congruence?

<p>A statement of congruence can be reversed.</p> Signup and view all the answers

What is the transitive property of congruence?

<p>In a &quot;chain&quot; of congruences, the first triangle is congruent to the last triangle.</p> Signup and view all the answers

You can build two triangles that have the same side lengths, but are not congruent.

<p>False</p> Signup and view all the answers

What is the SSS Postulate?

<p>The SSS Postulate states that if the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.</p> Signup and view all the answers

What is the SAS Postulate?

<p>The SAS Postulate states that 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 triangles are congruent.</p> Signup and view all the answers

What is the ASA Postulate?

<p>The ASA Postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.</p> Signup and view all the answers

What is the AAS Theorem?

<p>The AAS Theorem states that if two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and a nonincluded side of another triangle, then the triangles are congruent.</p> Signup and view all the answers

SSA (side-side-angle) guarantees congruence between two triangles.

<p>False</p> Signup and view all the answers

AAA (angle-angle-angle) guarantees congruence between two triangles.

<p>False</p> Signup and view all the answers

What are congruence postulates that do not work?

<p>SSA, AAA</p> Signup and view all the answers

What does "similar" mean?

<p>It is the term used in geometry to describe figures with the same shape.</p> Signup and view all the answers

What does dilate mean?

<p>Dilate means to change the scale of an object.</p> Signup and view all the answers

Dilating a triangle changes the ____ of the triangle but does not change its ____.

<p>size; shape</p> Signup and view all the answers

Similar triangles can be made into congruent triangles by ______ them.

<p>dilating</p> Signup and view all the answers

With similar triangles, the ratios of all three pairs of corresponding sides are ____.

<p>equal</p> Signup and view all the answers

Similar triangles are triangles whose corresponding angles are ________ and whose corresponding sides are ________ in length.

<p>congruent; proportional</p> Signup and view all the answers

What is a scale factor?

<p>It is the ratio of any side length to the corresponding side length of the original triangle.</p> Signup and view all the answers

What is a ratio?

<p>A ratio is the relation between two quantities expressed as the quotient of one divided by the other.</p> Signup and view all the answers

What is a proportion?

<p>A proportion is an equation that states that two ratios are equal.</p> Signup and view all the answers

What is a mean of a proportion?

<p>B,C; if a proportion was expressed as A/B = C/D</p> Signup and view all the answers

What is an extreme of a proportion?

<p>A,D; if a proportion was expressed as A/B = C/D</p> Signup and view all the answers

What does the Cross Product Property say about this proportion?

<p>It says that the product of the means equals the products of the extremes.</p> Signup and view all the answers

What is the AA Similarity Postulate?

<p>The AA Similarity Postulate states that if two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar.</p> Signup and view all the answers

What is the SSS Similarity Theorem?

<p>The SSS Similarity Theorem states that if the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.</p> Signup and view all the answers

What is the SAS Similarity Theorem?

<p>The SAS Similarity Theorem states that if an angle of one triangle is congruent to an angle of another triangle, and if the lengths of the sides including these angles are proportional, then the triangles are similar.</p> Signup and view all the answers

What is the Isosceles Triangle Theorem?

<p>The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent.</p> Signup and view all the answers

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

<p>True</p> Signup and view all the answers

What are the two corollaries of the Isosceles Triangle Theorem?

<ol> <li>Equilateral triangles are also equiangular, and vice versa. 2) Each angle of an equilateral triangle measures 60º</li> </ol> Signup and view all the answers

The longest side of a triangle is always opposite the angle with the _______ measure.

<p>largest</p> Signup and view all the answers

The shortest side of a triangle is always opposite the angle with the _______ measure.

<p>smallest</p> Signup and view all the answers

What is the median of a triangle?

<p>A line or segment that passes through a vertex and bisects the side opposite the vertex.</p> Signup and view all the answers

What is a centroid?

<p>The common point of intersection shared by the medians of a triangle.</p> Signup and view all the answers

What is an altitude of a triangle?

<p>A line or segment through a vertex and is perpendicular to the line containing the opposite side.</p> Signup and view all the answers

What is an orthocenter?

<p>The common point of intersection shared by the altitudes of a triangle.</p> Signup and view all the answers

What is an angle bisector?

<p>A line or segment that bisects a vertex and bisects the side opposite from it.</p> Signup and view all the answers

What is the incenter?

<p>The point shared by the triangle's angle bisectors.</p> Signup and view all the answers

What is a perpendicular bisector?

<p>A line or segment that bisects the side opposite from it at a 90º angle.</p> Signup and view all the answers

What is a circumcenter?

<p>The point shared by the perpendicular bisectors of the triangle's sides.</p> Signup and view all the answers

What is the center of gravity for the triangle?

<p>The centroid</p> Signup and view all the answers

Which may fall outside of a triangle?

<p>Circumcenter</p> Signup and view all the answers

Study Notes

Triangle Basics

  • A triangle is formed by three segments connecting three noncollinear points.
  • Parts of a triangle include three sides, three vertices, and three angles.
  • A triangle is named using the letters corresponding to its vertices.

Types of Triangles

  • Acute triangles have all angles less than 90º.
  • Right triangles contain exactly one right angle (90º).
  • Obtuse triangles have exactly one obtuse angle (greater than 90º).
  • Equilateral triangles have three congruent sides and angles.
  • Isosceles triangles have at least two congruent sides.
  • Scalene triangles have no sides of equal length.

Triangle Properties

  • The sum of all internal angles in a triangle is 180º.
  • The triangle inequality rule states that the sum of the lengths of any two sides must exceed the length of the third side.
  • Two triangles with two congruent angles each have congruent third angles.

External Angles and Congruence

  • A triangle has six exterior angles, corresponding to its three interior angles.
  • The sum of an exterior angle equals the sum of its two remote interior angles.
  • Congruent triangles share the same size and shape.

Triangle Congruence

  • Congruence transformations include translating, rotating, and reflecting.
  • Corresponding parts of congruent triangles are congruent (CPCTC).
  • The SSS Postulate establishes congruence by comparing all sides.
  • The SAS Postulate uses two sides and the included angle for congruence.
  • The ASA Postulate applies two angles and the included side for triangle congruence.
  • The AAS Theorem considers two angles and a non-included side to determine congruence.

Non-Working Congruence Postulates

  • SSA (side-side-angle) and AAA (angle-angle-angle) do not guarantee triangle congruence.

Similar Triangles

  • Similar triangles have the same shape with corresponding angles congruent and sides proportional.
  • Dilation changes the size of an object without altering its shape.
  • Similarity refers to figures that maintain the same shape.

Proportions and Ratios

  • Mean of a proportion consists of the middle terms in a proportion's equation.
  • Extremes of a proportion refer to the outer terms in its equation.
  • The Cross Product Property states that the product of the means equals the product of the extremes.

Triangle Theorems and Corollaries

  • AA Similarity Postulate states that two triangles are similar if two angles are congruent.
  • The SSS and SAS Similarity Theorems establish similarity based on proportions of corresponding side lengths or angles.
  • The Isosceles Triangle Theorem states angles opposite congruent sides are congruent.
  • Two corollaries of the Isosceles Triangle Theorem highlight that equilateral triangles are equiangular, and each angle measures 60º.

Triangle Centers and Segments

  • The longest side is opposite the largest angle, while the shortest side is opposite the smallest angle.
  • A median bisects the opposite side and connects to a vertex.
  • A centroid is the intersection point of a triangle's medians and acts as the triangle's center of gravity.
  • An altitude is perpendicular to the opposite side, meeting at a vertex.
  • The orthocenter is the intersection point of a triangle's altitudes.
  • An angle bisector divides a vertex's angle and bisects the opposite side.
  • The incenter is the intersection point of the angle bisectors.
  • A perpendicular bisector cuts a side at a right angle, intersecting at the circumcenter, which is formed by the perpendicular bisectors of the sides.

Special Cases in Triangle Centers

  • The orthocenter and circumcenter may be located outside of the triangle.

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