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Questions and Answers
In an isosceles trapezoid $ABCD$, where $AB$ and $CD$ are the bases, which of the following statements is NOT necessarily true?
In an isosceles trapezoid $ABCD$, where $AB$ and $CD$ are the bases, which of the following statements is NOT necessarily true?
Given a kite $EFGH$ where $EG$ is the longer diagonal and $FH$ is the shorter diagonal. If $∠FEH = 50°$, what is the measure of $∠FGH$?
Given a kite $EFGH$ where $EG$ is the longer diagonal and $FH$ is the shorter diagonal. If $∠FEH = 50°$, what is the measure of $∠FGH$?
A trapezoid has bases of length 10 and 14. What is the length of its median?
A trapezoid has bases of length 10 and 14. What is the length of its median?
Quadrilateral $PQRS$ has diagonals that are perpendicular bisectors of each other. Which of the following classifications is $PQRS$ guaranteed to be?
Quadrilateral $PQRS$ has diagonals that are perpendicular bisectors of each other. Which of the following classifications is $PQRS$ guaranteed to be?
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Two similar triangles, $ΔABC$ and $ΔDEF$, have corresponding sides $AB = 6$ and $DE = 9$. If the area of $ΔABC$ is 24 square units, what is the area of $ΔDEF$?
Two similar triangles, $ΔABC$ and $ΔDEF$, have corresponding sides $AB = 6$ and $DE = 9$. If the area of $ΔABC$ is 24 square units, what is the area of $ΔDEF$?
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In parallelogram $ABCD$, diagonals $AC$ and $BD$ intersect at point $E$. If $AE = 3x + 2$ and $EC = 7x - 6$, what is the length of $AC$?
In parallelogram $ABCD$, diagonals $AC$ and $BD$ intersect at point $E$. If $AE = 3x + 2$ and $EC = 7x - 6$, what is the length of $AC$?
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Which of the following conditions is sufficient to prove that a quadrilateral is a parallelogram?
Which of the following conditions is sufficient to prove that a quadrilateral is a parallelogram?
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In rectangle $PQRS$, $PR = 5x - 3$ and $QS = 3x + 5$. What is the length of $PR$?
In rectangle $PQRS$, $PR = 5x - 3$ and $QS = 3x + 5$. What is the length of $PR$?
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Which property is unique to rhombuses compared to all other parallelograms?
Which property is unique to rhombuses compared to all other parallelograms?
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In rhombus $WXYZ$, if $\angle WXY = 60^{\circ}$, what is the measure of $\angle WZY$?
In rhombus $WXYZ$, if $\angle WXY = 60^{\circ}$, what is the measure of $\angle WZY$?
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Three lines intersect at the same point. What is this configuration called?
Three lines intersect at the same point. What is this configuration called?
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If the consecutive angles of a quadrilateral are supplementary, which type of quadrilateral can it be definitively classified as?
If the consecutive angles of a quadrilateral are supplementary, which type of quadrilateral can it be definitively classified as?
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Which of the following statements is always true for a square?
Which of the following statements is always true for a square?
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Flashcards
Properties of a square
Properties of a square
All angles are right; diagonals bisect and are congruent.
Median of a trapezoid
Median of a trapezoid
A segment joining midpoints of non-parallel sides, half the sum of the bases.
Isosceles trapezoid
Isosceles trapezoid
A trapezoid with congruent legs and base angles; diagonals are also congruent.
Properties of a kite
Properties of a kite
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Similar figures
Similar figures
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Parallelogram
Parallelogram
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Properties of Parallelograms
Properties of Parallelograms
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Conditions for Parallelogram
Conditions for Parallelogram
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Rectangle
Rectangle
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Rhombus
Rhombus
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Square
Square
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Diagonals of Parallelogram
Diagonals of Parallelogram
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Supplementary Angles
Supplementary Angles
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Study Notes
Parallelograms
- Parallelograms are quadrilaterals with opposite sides parallel.
- Intersecting lines share a common point.
- Concurrent lines share a common point.
- Perpendicular lines intersect at a right angle.
- Transversal lines intersect two or more lines at distinct points.
Properties of Parallelograms
- Opposite sides are parallel and congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary (add up to 180 degrees).
- Diagonals bisect each other.
- Each diagonal divides the parallelogram into two congruent triangles.
Conditions for a Quadrilateral to be a Parallelogram
- Condition 1: If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
- Condition 2: If one pair of opposite sides is congruent and parallel, then the quadrilateral is a parallelogram.
- Condition 3: If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.
- Condition 4: If consecutive angles are supplementary, then the quadrilateral is a parallelogram.
- Condition 5: If the diagonals bisect each other, then the quadrilateral is a parallelogram.
- Condition 6: If each diagonal forms two congruent triangles, then the quadrilateral is a parallelogram.
Rectangles
- A parallelogram with four right angles.
- Equiangular but not necessarily equilateral.
- Opposite sides are parallel and congruent.
- All angles are congruent and are right angles.
- Consecutive angles are supplementary.
- Diagonals bisect each other and are congruent.
- Each diagonal divides the rectangle into two congruent triangles.
Rhombus
- A quadrilateral with four congruent sides.
- Equiangular but not equilateral.
- Opposite sides are parallel and congruent.
- All sides are congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary.
- Each diagonal bisects a pair of opposite angles.
- Each diagonal is a perpendicular bisector of the other diagonal.
- Each diagonal divides the rhombus into two congruent triangles.
- The diagonals of a rhombus are perpendicular.
Square
- A quadrilateral with four congruent sides and four congruent angles (right angles).
- All sides are parallel and congruent.
- All sides are congruent.
- All angles are congruent and are right angles.
- Opposite angles are congruent and supplementary.
- Consecutive angles are supplementary and congruent.
- Diagonals are congruent and bisect opposite angles.
- Each diagonal is a perpendicular bisector of the other diagonal.
- Each diagonal divides the square into two congruent triangles.
Trapezoid
- A quadrilateral with exactly one pair of parallel sides.
- The parallel sides are called bases.
- The non-parallel sides are called legs.
- The median of a trapezoid is a segment connecting the midpoints of the legs. It's parallel to the bases and its length is half the sum of the lengths of the bases.
- Isosceles Trapezoid: A trapezoid in which the legs are congruent. Base angles are congruent and diagonals are congruent.
Kite
- A quadrilateral with two distinct pairs of consecutive congruent sides.
- Only one pair of opposite angles is congruent.
- One diagonal bisects the other.
- The diagonals are perpendicular.
- The area of a kite is half the product of the lengths of its diagonals.
Similar Figures
- Figures with the same shape but not necessarily the same size.
- Corresponding sides are proportional.
- Corresponding angles are congruent.
Congruent Figures
- Figures with the same shape and size.
- Corresponding sides and angles are congruent.
Basic Proportionality Theorem (BPT)
- If a line is drawn parallel to one side of a triangle and intersects the other two sides in distinct points, then it divides the sides proportionally.
Right Triangle Similarity Theorem
- If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to each other and to the original right triangle.
Geometric Mean Theorem
- In a right triangle, the altitude to the hypotenuse divides the hypotenuse in two segments, and the altitude is the geometric mean of the segments. The length of a leg is the geometric mean of the hypotenuse and the adjacent segment.
Special Right Triangles
- 45-45-90 Triangles: The hypotenuse is √2 times the length of a leg.
- 30-60-90 Triangles: The hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times the shorter leg.
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Description
Explore the properties of parallelograms, including congruent sides and angles. Understand the conditions that define a quadrilateral as a parallelogram, such as congruent and parallel opposite sides. Learn about transversals, intersecting, concurrent, & perpendicular lines.
