Podcast
Questions and Answers
What is a triangle?
What is a triangle?
A figure formed by three segments connecting three noncollinear points.
What are the parts of a triangle?
What are the parts of a triangle?
Three sides, three vertices, and three angles.
How is a triangle named?
How is a triangle named?
A triangle is named by using the three letters that name its vertices.
What is an acute triangle?
What is an acute triangle?
What is a right triangle?
What is a right triangle?
What is an obtuse triangle?
What is an obtuse triangle?
What is an equilateral triangle?
What is an equilateral triangle?
What is an isosceles triangle?
What is an isosceles triangle?
Triangles that are equilateral are also isosceles.
Triangles that are equilateral are also isosceles.
What is a scalene triangle?
What is a scalene triangle?
What is the rule for what side lengths can form a triangle?
What is the rule for what side lengths can form a triangle?
What is the sum of all the angles in a triangle?
What is the sum of all the angles in a triangle?
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are ____.
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are ____.
How many exterior angles does a triangle have?
How many exterior angles does a triangle have?
What are remote interior angles?
What are remote interior angles?
What is the relationship between an exterior angle and its two remote interior angles?
What is the relationship between an exterior angle and its two remote interior angles?
Congruent triangles have exactly the same ____ and ____.
Congruent triangles have exactly the same ____ and ____.
The three congruence transformations are _______, _______, and _______.
The three congruence transformations are _______, _______, and _______.
How can you find the corresponding parts of two congruent triangles?
How can you find the corresponding parts of two congruent triangles?
If two triangles are congruent, all three pairs of corresponding _____ and all three pairs of corresponding _____ are also congruent.
If two triangles are congruent, all three pairs of corresponding _____ and all three pairs of corresponding _____ are also congruent.
What are congruent triangles?
What are congruent triangles?
What is CPCTC?
What is CPCTC?
What is the reflexive property of congruence?
What is the reflexive property of congruence?
What is the symmetric property of congruence?
What is the symmetric property of congruence?
What is the transitive property of congruence?
What is the transitive property of congruence?
You can build two triangles that have the same side lengths, but are not congruent.
You can build two triangles that have the same side lengths, but are not congruent.
What is the SSS Postulate?
What is the SSS Postulate?
What is the SAS Postulate?
What is the SAS Postulate?
What is the ASA Postulate?
What is the ASA Postulate?
What is the AAS Theorem?
What is the AAS Theorem?
SSA (side-side-angle) guarantees congruence between two triangles.
SSA (side-side-angle) guarantees congruence between two triangles.
AAA (angle-angle-angle) guarantees congruence between two triangles.
AAA (angle-angle-angle) guarantees congruence between two triangles.
What are congruence postulates that do not work?
What are congruence postulates that do not work?
What does "similar" mean?
What does "similar" mean?
What does dilate mean?
What does dilate mean?
Dilating a triangle changes the ____ of the triangle but does not change its ____.
Dilating a triangle changes the ____ of the triangle but does not change its ____.
Similar triangles can be made into congruent triangles by ______ them.
Similar triangles can be made into congruent triangles by ______ them.
With similar triangles, the ratios of all three pairs of corresponding sides are ____.
With similar triangles, the ratios of all three pairs of corresponding sides are ____.
Similar triangles are triangles whose corresponding angles are ________ and whose corresponding sides are ________ in length.
Similar triangles are triangles whose corresponding angles are ________ and whose corresponding sides are ________ in length.
What is a scale factor?
What is a scale factor?
What is a ratio?
What is a ratio?
What is a proportion?
What is a proportion?
What is a mean of a proportion?
What is a mean of a proportion?
What is an extreme of a proportion?
What is an extreme of a proportion?
What does the Cross Product Property say about this proportion?
What does the Cross Product Property say about this proportion?
What is the AA Similarity Postulate?
What is the AA Similarity Postulate?
What is the SSS Similarity Theorem?
What is the SSS Similarity Theorem?
What is the SAS Similarity Theorem?
What is the SAS Similarity Theorem?
What is the Isosceles Triangle Theorem?
What is the Isosceles Triangle Theorem?
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
What are the two corollaries of the Isosceles Triangle Theorem?
What are the two corollaries of the Isosceles Triangle Theorem?
The longest side of a triangle is always opposite the angle with the _______ measure.
The longest side of a triangle is always opposite the angle with the _______ measure.
The shortest side of a triangle is always opposite the angle with the _______ measure.
The shortest side of a triangle is always opposite the angle with the _______ measure.
What is the median of a triangle?
What is the median of a triangle?
What is a centroid?
What is a centroid?
What is an altitude of a triangle?
What is an altitude of a triangle?
What is an orthocenter?
What is an orthocenter?
What is an angle bisector?
What is an angle bisector?
What is the incenter?
What is the incenter?
What is a perpendicular bisector?
What is a perpendicular bisector?
What is a circumcenter?
What is a circumcenter?
What is the center of gravity for the triangle?
What is the center of gravity for the triangle?
Which may fall outside of a triangle?
Which may fall outside of a triangle?
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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