HON Geometry Topics 1-3 Study Guide Flashcards

Flashcard study set covering topics 1 through 3 in HON Geometry including line/angle relationships, triangle theorems, parallel lines, and transformations.

Equilateral Triangle

A triangle in which all three sides and all three interior angles are congruent ($60^\circ$ each).

Isosceles Triangle

A triangle with two congruent sides; the interior angles opposite these sides are also congruent.

Scalene Triangle

A triangle with no congruent sides or angles, where the largest angle opens up to the largest side.

Supplementary Angles

Two angles whose measures add up to $180^\circ$.

Complementary Angles

Two angles whose measures add up to $90^\circ$.

Linear Pair

A pair of adjacent angles whose non-common sides form a straight line, making them supplementary ($\text{sum} = 180^\circ$).

Vertical Angles

A pair of non-adjacent opposite angles formed by two intersecting lines; vertical angles are congruent.

Exterior Angle Theorem

The measure of an exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Corresponding Angles

Angles in matching positions relative to two lines and a transversal; they are congruent when the lines are parallel.

Alternate Interior Angles

Angles on opposite sides of the transversal and inside two parallel lines; alternate interior angles are congruent.

Alternate Exterior Angles

Angles on opposite sides of the transversal and outside two parallel lines; alternate exterior angles are congruent.

Consecutive Interior Angles

Angles on the same side of the transversal and inside two parallel lines; consecutive interior angles are supplementary ($\text{sum} = 180^\circ$).

Translation Rule

A transformation that slides every point of a figure by the same distance and direction, represented algebraically as $(x, y) \rightarrow (x + a, y + b)$.

Reflection across the x-axis

A geometric transformation that flips a figure over the x-axis, mapping coordinate points according to $(x, y) \rightarrow (x, -y)$.

Reflection across the y-axis

A geometric transformation that flips a figure over the y-axis, mapping coordinate points according to $(x, y) \rightarrow (-x, y)$.

Reflection across line y = x

A geometric transformation that flips a figure across the line $y = x$, mapping coordinate points according to $(x, y) \rightarrow (y, x)$.

Reflection across line y = -x

A geometric transformation that flips a figure across the line $y = -x$, mapping coordinate points according to $(x, y) \rightarrow (-y, -x)$.

Rotation $180^\circ$ about the origin

A transformation turning a figure $180^\circ$ around the origin, mapping coordinate points according to $(x, y) \rightarrow (-x, -y)$.

Rotation $90^\circ$ clockwise about the origin

A transformation turning a figure $90^\circ$ clockwise around the origin, mapping coordinate points according to $(x, y) \rightarrow (y, -x)$.

Rotation $90^\circ$ counterclockwise about the origin

A transformation turning a figure $90^\circ$ counterclockwise around the origin, mapping coordinate points according to $(x, y) \rightarrow (-y, x)$.

Problem 1 Solution (Value of $x$)

$x = 22$. The two angles form a right angle ($90^\circ$), so $3x + (x + 2) = 90$.

Problem 2 Solution (Value of $x$)

$x = 15$. The angles form a linear pair ($180^\circ$), so $87 + (6x + 3) = 180$.

Problem 3 Solution (Value of $x$)

$x = 41$. Vertical angles are congruent, so $2x + 1 = 83$.

Problem 6 Solution (Value of $x$)

$x = -4$. The sum of interior angles in a triangle is $180^\circ$, so $71 + 73 + (x + 40) = 180$.

Problem 7 Solution (Value of $x$)

$x = -4$. By the Exterior Angle Theorem, $(x + 129) = (180 - 105) + 50 = 125$.

Problem 8 Solution (Value of $x$)

$x = 7$. In an isosceles triangle, base angles are equal: $m\angle 2 = \frac{180 - 100}{2} = 40^\circ$. Solving $4x + 12 = 40$ gives $x = 7$.

Isosceles Base Angle Problem (Value of $x$)

$x = 11$. Base angles are congruent ($m\angle 2 = 43^\circ$). Solving $3x + 10 = 43$ yields $x = 11$.

Isosceles Triangle Side Problem (Value of $x$)

$x = 8$. Sides opposite congruent angles are equal, so $2x - 5 = 11$.

Isosceles Vertex Angle Problem (Value of $x$)

$x = 82$. The base angles are both $49^\circ$, so $x = 180 - 2(49) = 82$.

Problem 13 Solution (Value of $x$)

$x = 8$. Alternate interior angles are congruent, so $15x + 10 = 130$.

Problem 14 Solution (Value of $x$)

$x = 3$. Corresponding angles are congruent, so $13x - 4 = 8 + 9x$.

Supplementary Angle Difference Problem

Two supplementary angles that differ by $98^\circ$ measure $41^\circ$ and $139^\circ$.