Grid & Angle Quest

TEACHER ANSWER KEY & SOLUTION GUIDE

Name: SOLUTIONS KEY Date: CLASS KEY

Class: GEOMETRY PRE-AP Score: 100%

Pedagogical Tip: Use this key to trace student work step-by-step. Common misconceptions are noted below in red boxes. Encourage students to check arithmetic on signs.

Part 1: Transformation Station (Answers)

Q1.1 KEY Reflection

Identify the specific transformation that maps \(\Delta ABC\) to \(\Delta A'B'C'\).

Transformation Type:

Reflection (Horizontal flip)

Rule / Line of Symmetry:

Reflected across the y-axis (line \(x=0\))

Q1.2 KEY Translation

A translation maps point \(P(3, -4)\) to \(P'(7, -9)\). Find the algebraic rule for this translation, then apply it to translate \(Q(-1, 5)\).

Step 1: Write the General Rule

\( (x, y) \rightarrow ( x + 4, y - 5 ) \)

Check: \( 3+4=7 \) and \( -4-5=-9 \).

Step 2: Find Coordinates of \(Q'\)

\( Q' = ( -1 + 4, 5 - 5 ) = (3, 0) \)

Part 2: Angle Alliances (Answers)

Angle Pairs Identified

A) \(\angle 3\) & \(\angle 6\): Alternate Interior

B) \(\angle 2\) & \(\angle 3\): Vertical Angles

C) \(\angle 1\) & \(\angle 5\): Corresponding

D) \(\angle 4\) & \(\angle 5\): Consecutive Interior

Q2.1 KEY Angle Measures

Suppose \(m\angle 1 = 124^\circ\). Find the measure of each angle below and explain your geometric reasoning.

a) Find \(m\angle 4\):

\(m\angle 4 = 124^\circ\)

Geometric Reason:

Vertical angles are congruent (\(\angle 1 \cong \angle 4\)).

b) Find \(m\angle 8\):

\(m\angle 8 = 124^\circ\)

Geometric Reason:

Alternate Exterior Angles are congruent (\(\angle 1 \cong \angle 8\)).

Section A Answers Page 1 of 3

Grid & Angle Quest

TEACHER ANSWER KEY & SOLUTION GUIDE

Part 2 of 3

Part 3: Distance Detective (Answers)

Q3.1 KEY Distance Steps

Calculate the distance between points \(A(2, -1)\) and \(B(8, 7)\). Write down all coordinates clearly in the formula steps.

SOLUTION STEPS:

1. Identify: \( (x_1, y_1) = ( \mathbf{2}, \mathbf{-1} ) \) & \( (x_2, y_2) = ( \mathbf{8}, \mathbf{7} ) \)

2. Substitute: \( d = \sqrt{( \mathbf{8} - \mathbf{2} )^2 + ( \mathbf{7} - (\mathbf{-1}) )^2} \)

3. Simplify: \( d = \sqrt{ ( \mathbf{6} )^2 + ( \mathbf{8} )^2 } = \sqrt{36+64} = \sqrt{100} \)

4. Final Answer: \( d = \mathbf{10} \) units

Q3.2 KEY Distance Graph

Find the length of line segment \(CD\) shown on the grid below.

Pythagorean approach:
Horizontal leg \(\Delta x = 2 - (-2) = 4\)
Vertical leg \(\Delta y = -1 - 2 = -3\)
\(d = \sqrt{4^2 + (-3)^2} = \sqrt{16+9} = \mathbf{5}\)

Part 4: Triangle Triumphs (Congruence Criteria Key)

Case A KEY SSS

Postulate: SSS

Statement: \(\Delta ABC \cong \Delta\) DEF

Case B KEY SAS

Postulate: SAS

Statement: \(\Delta LMN \cong \Delta\) XYZ

Case C KEY ASA

Postulate: ASA

Statement: \(\Delta PQR \cong \Delta\) STU

Case D KEY AAS

Postulate: AAS

Statement: \(\Delta GHI \cong \Delta\) JKL

Section B Answers Page 2 of 3

Grid & Angle Quest

TEACHER ANSWER KEY & SOLUTION GUIDE

The Final Challenge KEY

Part 5: The Lost Jungle Temple Quest (Answers)

1 TEMPLE COORDINATES (DISTANCE FORMULA)

The archaeologist stands at Basecamp \(B(-3, 2)\) and needs to march to the Temple Gate \(T(5, -4)\). Compute the exact coordinate distance between these points.

Solution steps:
\( d = \sqrt{(5 - (-3))^2 + (-4 - 2)^2} = \sqrt{(8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = \mathbf{10} \)

Common error: forgetting that \(5 - (-3) = 8\) (not 2). Distance \(d = \mathbf{10}\) units

2 MIRROR REFLECTION (TRANSFORMATIONS)

To open the heavy gate, a laser beam at point \(L(4, 3)\) must be reflected over the \(y\)-axis to point \(L'\). What are the coordinates of \(L'\)? Afterwards, translate \(L'\) under \((x - 5, y + 2)\).

a) Coordinates of reflected point \(L'\):

\(L' = ( \mathbf{-4}, \mathbf{3} )\)

Rule: \((x,y) \rightarrow (-x,y)\)

b) Translation of \(L'(x - 5, y + 2)\):

\(L'' = ( \mathbf{-9}, \mathbf{5} )\)

Computation: \((-4-5, 3+2)\)

3 THE TRIANGLE KEYHOLES (CONGRUENCE CRITERIA)

The lock on the chest consists of two triangular slots, \(\Delta MNP\) and \(\Delta RST\). You measure and find that: \(MN \cong RS\), \(NP \cong ST\), and \(\angle N \cong \angle S\).

Congruence Postulate:

SAS (Side-Angle-Side)

Complete Statement:

\(\Delta MNP \cong \Delta \mathbf{RST}\)

Reason: Angle \(\angle N\) is directly between congruent sides \(MN\) and \(NP\), satisfying SAS.

Section C Answers Page 3 of 3

Grid and Angle Quest Answer Key

Grid and Angle Quest Answer Key

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