Question

1. in △ jkl, if m∠j > 90°, then ∠k and ∠l are _.

• acute
• complementary
• congruent
• obtuse

1. what is m∠cbd?

triangle diagram with a, b, d, c; angles at a: 60°, at d (between a and d): 80°, at c: 25°
m∠cbd = _°

1. if a series of rigid transformations maps △ qrs onto △ mpn, match each side of mpn with its length.

triangle qrs with sides: rq=8, qs=6, rs=7
mn 6
mp 7
pn 8

Explanation:

Question 2

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). In \(\triangle JKL\), \(m\angle J>90^\circ\), so \(m\angle K + m\angle L=180^\circ - m\angle J\).

Step2: Analyze angle measures

Since \(m\angle J>90^\circ\), \(180^\circ - m\angle J<90^\circ\), and each of \(\angle K\) and \(\angle L\) must be less than \(90^\circ\) (acute angles, as acute angles are less than \(90^\circ\); complementary would mean their sum is \(90^\circ\), congruent is not guaranteed, obtuse is greater than \(90^\circ\) which they can't be here).

Step1: Find \(\angle BDC\)

\(\angle ADB = 80^\circ\), so \(\angle BDC = 180^\circ - 80^\circ=100^\circ\) (linear pair).

Step2: Use triangle angle sum in \(\triangle CBD\)

In \(\triangle CBD\), angles sum to \(180^\circ\). Given \(\angle C = 25^\circ\) and \(\angle BDC = 100^\circ\), so \(m\angle CBD=180^\circ - 25^\circ - 100^\circ = 55^\circ\) (Wait, original handwritten answer was wrong. Let's recalculate: \(\angle BDC\) is supplementary to \(\angle ADB\): \(180 - 80 = 100\). Then in \(\triangle CBD\), angles are \(\angle C = 25\), \(\angle BDC = 100\), so \(\angle CBD = 180 - 25 - 100 = 55\).)

Step1: Recall rigid transformations

Rigid transformations (translation, rotation, reflection) preserve side lengths, so \(\triangle QRS\cong\triangle MPN\), meaning corresponding sides are equal.

Step2: Match sides

• \(MN\) corresponds to \(RS\) (length 7)
• \(MP\) corresponds to \(RQ\) (length 8)
• \(PN\) corresponds to \(QS\) (length 6)

Wait, let's check the triangle: \(RQ = 8\), \(RS = 7\), \(QS = 6\). So:

• \(MN\): length 7 (matches \(RS\))
• \(MP\): length 8 (matches \(RQ\))
• \(PN\): length 6 (matches \(QS\))

Answer:

Acute

Question 3