Theorems about Parallelograms - Geometry
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Find $x$ if an isosceles triangle has base angles $3x^\circ$ and vertex angle $36^\circ$.
Find $x$ if an isosceles triangle has base angles $3x^\circ$ and vertex angle $36^\circ$.
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$x=24$. $3x + 3x + 36 = 180$, so $6x = 144$ and $x = 24$.
$x=24$. $3x + 3x + 36 = 180$, so $6x = 144$ and $x = 24$.
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What must be true about points $D$ and $E$ to conclude $DE$ is a midsegment in $\triangle ABC$?
What must be true about points $D$ and $E$ to conclude $DE$ is a midsegment in $\triangle ABC$?
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$D$ and $E$ are midpoints of two sides of $\triangle ABC$. Midsegments require endpoints at midpoints of two triangle sides.
$D$ and $E$ are midpoints of two sides of $\triangle ABC$. Midsegments require endpoints at midpoints of two triangle sides.
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Identify the length of midsegment $DE$ if it is parallel to $BC$ and $BC=2k$.
Identify the length of midsegment $DE$ if it is parallel to $BC$ and $BC=2k$.
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$DE=k$. Midsegment length is half the parallel side: $\frac{1}{2}(2k) = k$.
$DE=k$. Midsegment length is half the parallel side: $\frac{1}{2}(2k) = k$.
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Find $BC$ if $DE$ is a midsegment parallel to $BC$ and $DE=\frac{3}{2}$.
Find $BC$ if $DE$ is a midsegment parallel to $BC$ and $DE=\frac{3}{2}$.
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$BC=3$. $BC = 2 \times DE = 2 \times \frac{3}{2} = 3$.
$BC=3$. $BC = 2 \times DE = 2 \times \frac{3}{2} = 3$.
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What is the required conclusion if $D$ and $E$ are midpoints of $AB$ and $AC$?
What is the required conclusion if $D$ and $E$ are midpoints of $AB$ and $AC$?
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$DE\parallel BC$ and $DE=\frac{1}{2}BC$. The Triangle Midsegment Theorem's required conclusions.
$DE\parallel BC$ and $DE=\frac{1}{2}BC$. The Triangle Midsegment Theorem's required conclusions.
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Which segment is a median if $M$ is the midpoint of $BC$ in $\triangle ABC$?
Which segment is a median if $M$ is the midpoint of $BC$ in $\triangle ABC$?
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$AM$. A median connects a vertex to the opposite side's midpoint.
$AM$. A median connects a vertex to the opposite side's midpoint.
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What is the point of concurrency of the medians, and what is its standard symbol?
What is the point of concurrency of the medians, and what is its standard symbol?
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Centroid, often labeled $G$. The medians' intersection point, commonly labeled $G$.
Centroid, often labeled $G$. The medians' intersection point, commonly labeled $G$.
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Find $GM$ if $G$ is the centroid on median $AM$ and $AM=15$.
Find $GM$ if $G$ is the centroid on median $AM$ and $AM=15$.
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$GM=5$. $GM = \frac{1}{3} \times AM = \frac{1}{3} \times 15 = 5$.
$GM=5$. $GM = \frac{1}{3} \times AM = \frac{1}{3} \times 15 = 5$.
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Find $AG$ if $G$ is the centroid on median $AM$ and $AM=15$.
Find $AG$ if $G$ is the centroid on median $AM$ and $AM=15$.
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$AG=10$. $AG = \frac{2}{3} \times AM = \frac{2}{3} \times 15 = 10$.
$AG=10$. $AG = \frac{2}{3} \times AM = \frac{2}{3} \times 15 = 10$.
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Find $AM$ if $G$ is the centroid on median $AM$ and $GM=7$.
Find $AM$ if $G$ is the centroid on median $AM$ and $GM=7$.
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$AM=21$. $AM = 3 \times GM = 3 \times 7 = 21$ from centroid ratio.
$AM=21$. $AM = 3 \times GM = 3 \times 7 = 21$ from centroid ratio.
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Identify the equal sides in $\triangle ABC$ if it is isosceles with base $BC$.
Identify the equal sides in $\triangle ABC$ if it is isosceles with base $BC$.
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$AB\cong AC$. In an isosceles triangle, the two non-base sides are equal.
$AB\cong AC$. In an isosceles triangle, the two non-base sides are equal.
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Identify the base angles in $\triangle ABC$ if it is isosceles with base $BC$.
Identify the base angles in $\triangle ABC$ if it is isosceles with base $BC$.
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$\angle B$ and $\angle C$. Base angles are opposite the congruent sides in isosceles triangles.
$\angle B$ and $\angle C$. Base angles are opposite the congruent sides in isosceles triangles.
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What is the third angle if a triangle has angles $x^\circ$, $x^\circ$, and $110^\circ$?
What is the third angle if a triangle has angles $x^\circ$, $x^\circ$, and $110^\circ$?
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$x=35$. $x + x + 110 = 180$, so $2x = 70$ and $x = 35$.
$x=35$. $x + x + 110 = 180$, so $2x = 70$ and $x = 35$.
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What is the point of concurrency of the three medians of a triangle called?
What is the point of concurrency of the three medians of a triangle called?
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Centroid. The intersection point of all three medians in a triangle.
Centroid. The intersection point of all three medians in a triangle.
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What is a median of a triangle?
What is a median of a triangle?
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A segment from a vertex to the midpoint of the opposite side. A median connects a vertex to the opposite side's midpoint.
A segment from a vertex to the midpoint of the opposite side. A median connects a vertex to the opposite side's midpoint.
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Identify the line parallel to $BC$ if $D$ is midpoint of $AB$ and $E$ is midpoint of $AC$.
Identify the line parallel to $BC$ if $D$ is midpoint of $AB$ and $E$ is midpoint of $AC$.
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$DE\parallel BC$. The midsegment $DE$ is parallel to the third side $BC$.
$DE\parallel BC$. The midsegment $DE$ is parallel to the third side $BC$.
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Find $BC$ if $D,E$ are midpoints and the midsegment $DE=11$.
Find $BC$ if $D,E$ are midpoints and the midsegment $DE=11$.
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$BC=22$. Third side is twice the midsegment: $11 × 2 = 22$.
$BC=22$. Third side is twice the midsegment: $11 × 2 = 22$.
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Find $DE$ if $D,E$ are midpoints of two sides and $BC=18$.
Find $DE$ if $D,E$ are midpoints of two sides and $BC=18$.
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$DE=9$. Midsegment length is half of the third side: $18 ÷ 2 = 9$.
$DE=9$. Midsegment length is half of the third side: $18 ÷ 2 = 9$.
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What is the length relationship in $\triangle ABC$ if $D,E$ are midpoints of $AB,AC$?
What is the length relationship in $\triangle ABC$ if $D,E$ are midpoints of $AB,AC$?
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$DE=\frac{1}{2}BC$. Midsegments are exactly half the length of the third side.
$DE=\frac{1}{2}BC$. Midsegments are exactly half the length of the third side.
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What is the relationship between a midsegment $DE$ and the third side $BC$ in $\triangle ABC$?
What is the relationship between a midsegment $DE$ and the third side $BC$ in $\triangle ABC$?
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$DE\parallel BC$. Midsegments are always parallel to the third side.
$DE\parallel BC$. Midsegments are always parallel to the third side.
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What does the Triangle Midsegment Theorem conclude about the midsegment and the third side?
What does the Triangle Midsegment Theorem conclude about the midsegment and the third side?
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Midsegment is parallel and half as long as the third side. The Triangle Midsegment Theorem's two main conclusions.
Midsegment is parallel and half as long as the third side. The Triangle Midsegment Theorem's two main conclusions.
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What is the name of the segment joining the midpoints of two sides of a triangle?
What is the name of the segment joining the midpoints of two sides of a triangle?
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Midsegment. The segment connecting midpoints of two triangle sides.
Midsegment. The segment connecting midpoints of two triangle sides.
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Identify the congruent sides if $\angle B\cong\angle C$ in $\triangle ABC$.
Identify the congruent sides if $\angle B\cong\angle C$ in $\triangle ABC$.
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$AB\cong AC$. Congruent angles indicate the opposite sides are congruent.
$AB\cong AC$. Congruent angles indicate the opposite sides are congruent.
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Identify the congruent angles if $AB\cong AC$ in $\triangle ABC$.
Identify the congruent angles if $AB\cong AC$ in $\triangle ABC$.
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$\angle B\cong\angle C$. Congruent sides $AB \cong AC$ create congruent base angles.
$\angle B\cong\angle C$. Congruent sides $AB \cong AC$ create congruent base angles.
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What theorem states that if two angles of a triangle are congruent, then the opposite sides are congruent?
What theorem states that if two angles of a triangle are congruent, then the opposite sides are congruent?
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Converse of the Isosceles Triangle Theorem. The reverse of the Isosceles Triangle Theorem.
Converse of the Isosceles Triangle Theorem. The reverse of the Isosceles Triangle Theorem.
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What theorem states that if two sides of a triangle are congruent, then the base angles are congruent?
What theorem states that if two sides of a triangle are congruent, then the base angles are congruent?
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Isosceles Triangle Theorem. This theorem connects congruent sides to congruent angles.
Isosceles Triangle Theorem. This theorem connects congruent sides to congruent angles.
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What is the measure of the vertex angle in an isosceles triangle with base angles $32^\circ$?
What is the measure of the vertex angle in an isosceles triangle with base angles $32^\circ$?
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$116^\circ$. Vertex angle: $180° - 32° - 32° = 116°$ using angle sum.
$116^\circ$. Vertex angle: $180° - 32° - 32° = 116°$ using angle sum.
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What is the measure of each base angle in an isosceles triangle with vertex angle $40^\circ$?
What is the measure of each base angle in an isosceles triangle with vertex angle $40^\circ$?
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$70^\circ$. Base angles are equal: $(180° - 40°) ÷ 2 = 70°$ each.
$70^\circ$. Base angles are equal: $(180° - 40°) ÷ 2 = 70°$ each.
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What is the measure of the third angle if two angles are $95^\circ$ and $40^\circ$?
What is the measure of the third angle if two angles are $95^\circ$ and $40^\circ$?
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$45^\circ$. Use $180° - 95° - 40° = 45°$ from the angle sum theorem.
$45^\circ$. Use $180° - 95° - 40° = 45°$ from the angle sum theorem.
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What is the measure of the third angle if two angles are $50^\circ$ and $60^\circ$?
What is the measure of the third angle if two angles are $50^\circ$ and $60^\circ$?
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$70^\circ$. Use $180° - 50° - 60° = 70°$ from the angle sum theorem.
$70^\circ$. Use $180° - 50° - 60° = 70°$ from the angle sum theorem.
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