Theorems about Triangles - Geometry
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What is the definition of a transversal in geometry?
What is the definition of a transversal in geometry?
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A line that intersects two or more lines at distinct points. Creates angle pairs at each intersection point.
A line that intersects two or more lines at distinct points. Creates angle pairs at each intersection point.
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What property justifies adding the same value to both sides of an equation in proofs?
What property justifies adding the same value to both sides of an equation in proofs?
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Addition Property of Equality. Maintains equality when adding same values.
Addition Property of Equality. Maintains equality when adding same values.
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What is the definition of same-side (consecutive) interior angles?
What is the definition of same-side (consecutive) interior angles?
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Interior angles on the same side of the transversal. Between the lines on the same side.
Interior angles on the same side of the transversal. Between the lines on the same side.
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What is the definition of corresponding angles for two lines cut by a transversal?
What is the definition of corresponding angles for two lines cut by a transversal?
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Angles in matching positions at the two intersections. Same relative position at both intersection points.
Angles in matching positions at the two intersections. Same relative position at both intersection points.
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Find $m\angle 2$ if $m\angle 1 = 115^\circ$ and $\angle 1$ and $\angle 2$ are alternate interior with parallel lines.
Find $m\angle 2$ if $m\angle 1 = 115^\circ$ and $\angle 1$ and $\angle 2$ are alternate interior with parallel lines.
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$m\angle 2 = 115^\circ$. Alternate interior angles are congruent with parallel lines.
$m\angle 2 = 115^\circ$. Alternate interior angles are congruent with parallel lines.
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What property justifies adding the same value to both sides of an equation in proofs?
What property justifies adding the same value to both sides of an equation in proofs?
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Addition Property of Equality. Maintains equality when adding same values.
Addition Property of Equality. Maintains equality when adding same values.
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What is the relationship between adjacent angles that form a linear pair?
What is the relationship between adjacent angles that form a linear pair?
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They are supplementary (sum to $180^\circ$). Adjacent angles forming a straight line sum to $180^\circ$.
They are supplementary (sum to $180^\circ$). Adjacent angles forming a straight line sum to $180^\circ$.
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What is the measure relationship for supplementary angles $m\angle A$ and $m\angle B$?
What is the measure relationship for supplementary angles $m\angle A$ and $m\angle B$?
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$m\angle A + m\angle B = 180^\circ$. Two angles whose measures add to $180^\circ$.
$m\angle A + m\angle B = 180^\circ$. Two angles whose measures add to $180^\circ$.
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What is the measure relationship for complementary angles $m\angle A$ and $m\angle B$?
What is the measure relationship for complementary angles $m\angle A$ and $m\angle B$?
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$m\angle A + m\angle B = 90^\circ$. Two angles whose measures add to $90^\circ$.
$m\angle A + m\angle B = 90^\circ$. Two angles whose measures add to $90^\circ$.
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Which postulate justifies splitting an angle into two parts to add measures?
Which postulate justifies splitting an angle into two parts to add measures?
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Angle Addition Postulate. States that $m\angle ABC = m\angle ABD + m\angle DBC$.
Angle Addition Postulate. States that $m\angle ABC = m\angle ABD + m\angle DBC$.
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What property justifies that if $a=b$ then $b=a$ in an angle proof?
What property justifies that if $a=b$ then $b=a$ in an angle proof?
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Symmetric Property of Equality. Allows reversing the order of an equality.
Symmetric Property of Equality. Allows reversing the order of an equality.
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What property justifies that if $a=b$ and $b=c$ then $a=c$ in an angle proof?
What property justifies that if $a=b$ and $b=c$ then $a=c$ in an angle proof?
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Transitive Property of Equality. Allows chaining equalities together.
Transitive Property of Equality. Allows chaining equalities together.
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What is the definition of alternate interior angles for two lines cut by a transversal?
What is the definition of alternate interior angles for two lines cut by a transversal?
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Interior angles on opposite sides of the transversal. Between the lines but on opposite sides.
Interior angles on opposite sides of the transversal. Between the lines but on opposite sides.
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What is the definition of alternate exterior angles for two lines cut by a transversal?
What is the definition of alternate exterior angles for two lines cut by a transversal?
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Exterior angles on opposite sides of the transversal. Outside the lines but on opposite sides.
Exterior angles on opposite sides of the transversal. Outside the lines but on opposite sides.
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What property justifies subtracting the same value from both sides of an equation in proofs?
What property justifies subtracting the same value from both sides of an equation in proofs?
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Subtraction Property of Equality. Maintains equality when subtracting same values.
Subtraction Property of Equality. Maintains equality when subtracting same values.
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What is the measure of each angle if two vertical angles together sum to $180^\circ$?
What is the measure of each angle if two vertical angles together sum to $180^\circ$?
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Each angle is $90^\circ$. Vertical angles sum to $180^\circ$ only when both are $90^\circ$.
Each angle is $90^\circ$. Vertical angles sum to $180^\circ$ only when both are $90^\circ$.
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Find $m\angle 1$ if $\angle 1$ and $\angle 2$ are vertical and $m\angle 2 = 3m\angle 1$.
Find $m\angle 1$ if $\angle 1$ and $\angle 2$ are vertical and $m\angle 2 = 3m\angle 1$.
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$m\angle 1 = 45^\circ$. Set $m\angle 1 + 3m\angle 1 = 180$, solve for $m\angle 1$.
$m\angle 1 = 45^\circ$. Set $m\angle 1 + 3m\angle 1 = 180$, solve for $m\angle 1$.
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Find $m\angle 2$ if $\angle 1$ and $\angle 2$ are a linear pair and $m\angle 2 = 2m\angle 1$.
Find $m\angle 2$ if $\angle 1$ and $\angle 2$ are a linear pair and $m\angle 2 = 2m\angle 1$.
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$m\angle 2 = 120^\circ$. Set $m\angle 1 + 2m\angle 1 = 180$, then $m\angle 2 = 2(60) = 120$.
$m\angle 2 = 120^\circ$. Set $m\angle 1 + 2m\angle 1 = 180$, then $m\angle 2 = 2(60) = 120$.
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What is the definition of vertical angles formed by two intersecting lines?
What is the definition of vertical angles formed by two intersecting lines?
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Opposite (nonadjacent) angles formed by intersecting lines. These angles share only a vertex and don't overlap.
Opposite (nonadjacent) angles formed by intersecting lines. These angles share only a vertex and don't overlap.
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Identify the correct conclusion if $PA = PB$ for point $P$ and segment $\overline{AB}$.
Identify the correct conclusion if $PA = PB$ for point $P$ and segment $\overline{AB}$.
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$P$ lies on the perpendicular bisector of $\overline{AB}$. Converse of perpendicular bisector theorem.
$P$ lies on the perpendicular bisector of $\overline{AB}$. Converse of perpendicular bisector theorem.
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Identify the correct conclusion if $P$ is on the perpendicular bisector of $\overline{AB}$.
Identify the correct conclusion if $P$ is on the perpendicular bisector of $\overline{AB}$.
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$PA = PB$. Direct application of perpendicular bisector theorem.
$PA = PB$. Direct application of perpendicular bisector theorem.
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Find $x$ if $PA = 2x+1$, $PB = 5x-11$, and $P$ lies on the perpendicular bisector of $\overline{AB}$.
Find $x$ if $PA = 2x+1$, $PB = 5x-11$, and $P$ lies on the perpendicular bisector of $\overline{AB}$.
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$x = 4$. Set equal distances: $2x+1=5x-11$, solve for $x$.
$x = 4$. Set equal distances: $2x+1=5x-11$, solve for $x$.
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Identify whether $P$ lies on the perpendicular bisector of $\overline{AB}$ if $PA = PB$.
Identify whether $P$ lies on the perpendicular bisector of $\overline{AB}$ if $PA = PB$.
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Yes; $PA=PB$ implies $P$ is on the perpendicular bisector. Converse of perpendicular bisector theorem applies.
Yes; $PA=PB$ implies $P$ is on the perpendicular bisector. Converse of perpendicular bisector theorem applies.
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Find $PB$ if $P$ lies on the perpendicular bisector of $\overline{AB}$ and $PA = 9$.
Find $PB$ if $P$ lies on the perpendicular bisector of $\overline{AB}$ and $PA = 9$.
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$PB = 9$. Perpendicular bisector theorem: equal distances to endpoints.
$PB = 9$. Perpendicular bisector theorem: equal distances to endpoints.
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Identify whether lines are parallel if same-side interior angles measure $120^\circ$ and $60^\circ$.
Identify whether lines are parallel if same-side interior angles measure $120^\circ$ and $60^\circ$.
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Yes; supplementary same-side interior angles imply parallel lines. Same-side interior angles sum to $180^\circ$.
Yes; supplementary same-side interior angles imply parallel lines. Same-side interior angles sum to $180^\circ$.
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Identify whether lines are parallel if alternate interior angles measure $48^\circ$ and $48^\circ$.
Identify whether lines are parallel if alternate interior angles measure $48^\circ$ and $48^\circ$.
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Yes; congruent alternate interior angles imply parallel lines. Equal alternate interior angles prove lines are parallel.
Yes; congruent alternate interior angles imply parallel lines. Equal alternate interior angles prove lines are parallel.
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Identify whether lines are parallel if corresponding angles measure $70^\circ$ and $70^\circ$.
Identify whether lines are parallel if corresponding angles measure $70^\circ$ and $70^\circ$.
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Yes; congruent corresponding angles imply parallel lines. Equal corresponding angles prove lines are parallel.
Yes; congruent corresponding angles imply parallel lines. Equal corresponding angles prove lines are parallel.
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Find $x$ if $m\angle 1 = 9x+9$ and $m\angle 2 = 3x+15$, and $\angle 1$ and $\angle 2$ are same-side interior with parallel lines.
Find $x$ if $m\angle 1 = 9x+9$ and $m\angle 2 = 3x+15$, and $\angle 1$ and $\angle 2$ are same-side interior with parallel lines.
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$x = 13$. Set same-side interior sum to $180$: $9x+9+3x+15=180$.
$x = 13$. Set same-side interior sum to $180$: $9x+9+3x+15=180$.
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Find $m\angle 2$ if $m\angle 1 = 101^\circ$ and $\angle 1$ and $\angle 2$ are same-side interior with parallel lines.
Find $m\angle 2$ if $m\angle 1 = 101^\circ$ and $\angle 1$ and $\angle 2$ are same-side interior with parallel lines.
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$m\angle 2 = 79^\circ$. Same-side interior angles are supplementary: $180-101=79$.
$m\angle 2 = 79^\circ$. Same-side interior angles are supplementary: $180-101=79$.
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Find $m\angle 2$ if $m\angle 1 = 115^\circ$ and $\angle 1$ and $\angle 2$ are alternate interior with parallel lines.
Find $m\angle 2$ if $m\angle 1 = 115^\circ$ and $\angle 1$ and $\angle 2$ are alternate interior with parallel lines.
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$m\angle 2 = 115^\circ$. Alternate interior angles are congruent with parallel lines.
$m\angle 2 = 115^\circ$. Alternate interior angles are congruent with parallel lines.
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