Category 1: Geometric Structure

The rectangular pyramid shown above was intersected by a plane parallel to base ABCD to form quadrilateral A'B'C'D'.

Based on this information, which statement cannot be proved true?

• $ABCD\ \sim\ A'B'C'D'$
  

• $\frac{AF}{A'F}=\frac{BF}{B'F}$

• $\angle AA'D'\cong\angle BB'A'$

• $\angle BCD\cong\angle B'C'D'$

For triangles ABC and DEF, ∠A ≅ ∠D and B ≅ ∠E. Based on this information, which statement is a reasonable conclusion?

• ∠C ≅ ∠D because they are corresponding angles of congruent triangles.

• CA ≅ FD because they are corresponding parts of congruent triangles.

• ∠C ≅ ∠F because they are corresponding angles of similar triangles.

• AB ≅ DE because they are corresponding parts of similar triangles.

A conditional statement is given above.

Which of the following best describes this statement?

• This statement is true because all obtuse triangles have two acute interior angles.

• This statement is false because the third interior angle must also be acute.

• This statement is true because a triangle can have at most one interior obtuse angle.

• This statement is false because the third interior angle can be acute, right, or obtuse.

The two conditional statements above are true.

Based on these conditional statements, which statement must also be true?

• If ∠3 and ∠4 form a linear pair, then

  m∠3 + m∠4 = 180^o

• If ∠3 and ∠4 form a linear pair, then

  m∠3 = 90^o, and m∠4 = 90^o

• If m∠3 + m∠4 = 180^o, then ∠3 and ∠4 form a linear pair

• If ∠3 and ∠4 are supplementary, then ∠3 and ∠4 form a linear pair

A statement is given above.

Which side length of a square provides a counterexample to the given statement?

• 6 units

• 4 units

• 10 units

• 2 units

In circle E above, ∠AEB ≌ ∠CED.

Based on this information, which statement must be true?

• AB ≌ CD

• EB ⊥ EC

• m∠AED = 4(m∠DEC)

• m∠AEB + m∠DEC = m∠BEC

The following conditional statement is true.

Which statement must also be true?

• If a quadrilateral has four congruent sides, then it is a square.

• If a quadrilateral does not have four congruent sides, then it is not a square.

• If a quadrilateral is not a square, then it does not have four congruent sides.

• If a quadrilateral does not have four congruent sides, then it is a square.

In the figure above, k ∥ t and k ⊥ q.

Based on this information, which statement can be proved true?

• ∠ACB ≌ ∠ABC

• ΔCAB is an acute triangle.

• ∠DCB ≌ ∠BCA

• ΔCAB is a right triangle.

The diagram above shows the arcs and segments used to construct ΔABC, given line k.

Based on this construction, which statement is not true?

• ∠AWC is complementary to ∠CWB.

• ΔCWB is a right triangle.

• ΔACB is isosceles.

• m∠CAB = m∠CBA

A town wants to fence in a rectangular section of a park. The table shows give possible plans for the dimensions of this fenced section. The changes in the width and the length of these plants follow a pattern.

If six additional plans are added to the table and follow the same pattern, which conclusion is not correct?

• The area of one of the additional plans exceeds 624 square feet.

• The area of one of the additional plans is less than 544 square feet.

• The area in Plan 6 is the same as the area in Plan 5.

• The area in Plan 7 is less than the area of Plan 6.