Questions and Answers
What are congruent polygons?
What does it mean to name a polygon?
To list the vertices in order going around the figure.
If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent.
True
You need to know the angle measures of a triangle to make a copy of it.
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What is triangle rigidity?
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The SSS postulate states that if three sides in one triangle are congruent to three sides in another triangle, then the triangles are congruent.
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The AAA combination establishes triangle congruence.
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In an isosceles triangle, the angles opposite the congruent sides are also congruent.
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What type of triangle has all three sides congruent?
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What is the measure of each angle in an equilateral triangle?
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If two angles of a triangle are congruent, what can be said about the sides opposite those angles?
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What is the property of the diagonals in a rhombus?
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If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
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What does the median from the vertex to the base of an isosceles triangle do?
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Study Notes
Congruent Polygons
- Congruent polygons have a one-to-one correspondence between their sides and angles.
- All pairs of corresponding angles and sides must be congruent.
Naming Polygons
- Polygons are named by listing their vertices in order, starting from any vertex, and can be done clockwise or counterclockwise.
Triangle Congruence
- Triangle congruence is established when the sides of one triangle are congruent to the corresponding sides of another.
Triangle Construction
- Knowing the measures of sides is sufficient to create a congruent triangle without needing angle measures.
Triangle Rigidity
- Fixed side lengths in a triangle lead to a unique shape, making triangles rigid.
Triangle Congruence Postulates
- SSS Postulate: Triangles are congruent if all three sides of one are congruent to the three sides of another.
- SAS Postulate: Triangles are congruent if two sides and the included angle in one triangle are congruent to the corresponding parts in another.
- ASA Postulate: Triangles are congruent if two angles and the included side in one triangle are congruent to the corresponding parts in another.
Non-Congruence Combinations
- AAA Combination: Three angles being congruent does not imply triangle congruence.
- SSA Combination: Two sides and a non-included angle do not establish triangle congruence.
AAS Theorem
- A triangle is congruent if two angles and a side not between them are congruent to another triangle's corresponding parts.
Postulate vs. Theorem
- Postulates are assumed true without proof, while theorems are proven true.
HL Theorem
- In right triangles, if the hypotenuse and one leg are congruent to another right triangle's corresponding parts, the triangles are congruent.
Congruent Triangles Definition
- Congruent triangles have corresponding parts that are congruent.
Isosceles Triangles
- Isosceles triangles have at least two congruent sides (legs) and a base, with a vertex angle opposite the base.
Special Properties of Isosceles Triangles
- The Isosceles Triangle Theorem states that if two sides are congruent, the angles opposite those sides are congruent.
- The converse states that if two angles are congruent, their opposite sides are congruent.
Properties of Equilateral Triangles
- Equilateral triangles have all sides congruent and each angle measures 60 degrees.
Bisectors and Medians in Isosceles Triangles
- The vertex angle bisector is also the perpendicular bisector of the base.
- The median to the base divides the triangle into two congruent triangles.
Parallelogram Properties
- Opposite sides and angles of a parallelogram are congruent.
- A diagonal divides a parallelogram into two congruent triangles.
Rhombus Properties
- A rhombus has all sides congruent, with diagonals forming four congruent triangles, being perpendicular, and congruent.
Rectangle Properties
- The diagonals of a rectangle are congruent. If the diagonals of a parallelogram are congruent, it is a rectangle.
Quadrilateral Criteria
- Various theorems establish conditions for quadrilaterals to be parallelograms based on properties of sides, angles, and diagonals.
Triangle Midsegment Theorem
- A segment joining midpoints of two sides of a triangle is parallel to the third side and half its length.
Congruent Radii Theorem
- All radii in the same or congruent circles are congruent.
Segment Addition and Triangle Inequality Postulates
- Segment addition states that if the sum of segments equals the whole, one point lies between the others.
- The triangle inequality postulate asserts that the sum of any two sides of a triangle must exceed the length of the third side.
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Description
Test your knowledge of key concepts in Geometry Chapter 4 with these flashcards. This quiz covers topics such as congruent polygons and how to name polygons effectively. Prepare for your exams by mastering these essential definitions and principles.
