- Use the vertical angles theorem to solve for \(x = 40^{\circ}\) (Option B).
- Identify the square as the quadrilateral with equal, bisecting, and perpendicular diagonals (Option C).
- Recognize AAA as a similarity, not a congruence condition (Option A).
- Apply the coordinate reflection rule \((x, y) \to (x, -y)\) to find \(P'(-3, -4)\) (Option B).
- Convert \(2.4 \text{ kL}\) to \(2,400,000 \text{ cm}^3\) using standard volume conversion factors (Option D).
Explanation
- Solving for x with Vertical Angles
- Vertical Angles Property: Two straight lines \(PQ\) and \(RS\) intersect at point \(T\), creating pairs of vertical angles. Vertical angles are always equal to each other.
- Equation: We are given \(\angle PTR = 2x + 30^{\circ}\) and \(\angle QTS = 110^{\circ}\). Since \(\angle PTR\) and \(\angle QTS\) are vertical angles, they must be equal:
\[2x + 30 = 110\]
- Solve for x: Subtract \(30\) from both sides:
\[2x = 110 - 30\]
\[2x = 80\]
\[x = 40\]
The value of \(x\) is \(40^{\circ}\), so the correct option is B.\
- Identify the Quadrilateral with Specific Diagonal Properties
- Quadrilateral Properties:
- Rectangle: Diagonals are equal and bisect each other, but they are not necessarily perpendicular.
- Rhombus: Diagonals bisect each other at right angles, but they are not necessarily equal in length.
- Square: Diagonals are equal in length, they bisect each other, and they intersect at right angles (perpendicular).
- Kite: Diagonals are perpendicular, but only one is bisected, and they are not equal.
- Conclusion: The square is the only quadrilateral that satisfies all three conditions: diagonals are equal, they bisect each other, and they are perpendicular. The correct option is C.
- Identifying the Incorrect Triangle Congruence Condition
- Congruence Criteria: Two triangles are congruent if they have the same shape and size. The recognized conditions for congruence are:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- RHS (Right angle-Hypotenuse-Side)
- Analysis: The AAA (Angle-Angle-Angle) condition proves that two triangles are similar (the same shape), but it does not guarantee they are the same size (congruent). Therefore, AAA is not a condition for congruence. The correct option is A.
- Reflecting a Point in the x-axis
- Reflection Rule: Reflecting a point \((x, y)\) in the x-axis changes the sign of the y-coordinate while the x-coordinate remains the same:
\[(x, y) \rightarrow (x, -y)\]
- Application: For the point \(P(-3, 4)\), applying the reflection rule gives:
\[(-3, 4) \rightarrow (-3, -4)\]
The coordinates of the image \(P'\) are \((-3; -4)\), so the correct option is B.
- Unit Conversion for Water Tank Volume
- Conversion Factors:
- \(1 \text{ kiloliter (kL)} = 1000 \text{ liters (L)}\)
- \(1 \text{ liter (L)} = 1000 \text{ cubic centimeters (cm}^3\text{)}\)
- Calculation:
- Convert \(2.4 \text{ kL}\) to \(\text{L}\): \(2.4 \times 1000 = 2400 \text{ L}\).
- Convert \(2400 \text{ L}\) to \(\text{cm}^3\): \(2400 \times 1000 = 2,400,000 \text{ cm}^3\).
The correct option is D.
Examples
Understanding these fundamental geometry and conversion concepts is essential in practical life. For example, recognizing vertical angles helps in surveying and construction layout. Knowing that a square's diagonals meet at right angles ensures perfect symmetry when installing tiles or building support structures. Reflection symmetry is used in computer graphics and design, while unit conversions are vital when buying materials or managing water usage in household tanks. Mastering these concepts ensures accuracy and efficiency in various real-world tasks.