KLMN is a parallelogram with KN//LM and LK//MN. Using the diagram above, prove the theorem which states that the opposite sides of a parallelogram are equal.

Since $\sin x$ is positive, $x$ is in Quadrant I or Quadrant II.
For Quadrant I:

For Quadrant II:

2.1.2 $\frac{1.53}{\tan(x-15^\circ)} = 1$

Since $\tan$ is positive, the angle is in Quadrant I or Quadrant III. The general solution for $\tan$ covers both.

For $k=0$:

For $k=1$:

2.2 Evaluate without using a calculator: $\frac{\cos 45^\circ}{\sin 45^\circ \left(\frac{1}{\cot^2 30^\circ}\right)}$

Cancel out the common term $\frac{\sqrt{2}}{2}$:

2.3 If $13 \cos \theta + 12 = 0$ and $180^\circ \le \theta \le 360^\circ$, calculate with the aid of diagram, but without using a calculator, the value of: $\frac{1}{\sec \theta} + \sin \theta$

Since $\cos \theta$ is negative and $180^\circ \le \theta \le 360^\circ$, $\theta$ must be in Quadrant III.

In Quadrant III, $x$ is negative, $y$ is negative, and $r$ is positive.
We have $\cos \theta = \frac{x}{r} = -\frac{12}{13}$. So, $x = -12$ and $r = 13$.
Use the Pythagorean theorem to find $y$:

Since $y$ is negative in Quadrant III, $y = -5$.

Recall that $\frac{1}{\sec \theta} = \cos \theta$.
So, $\frac{1}{\sec \theta} = -\frac{12}{13}$.

QUESTION 3

3.1 KLMN is a parallelogram with KN || LM and LK || MN. Using the diagram above, prove the theorem which states that the opposite sides of a parallelogram are equal.

Given: KLMN is a parallelogram.
This implies that KN || LM and KL || NM.

To Prove: KL = NM and KN = LM.

Construction: Draw diagonal LN.

Proof:
Consider $\triangle KLN$ and $\triangle MNL$.

1. $\angle KLN = \angle MNL$ (Alternate interior angles, since KL || NM and LN is a transversal).
2. $\angle KNL = \angle MLN$ (Alternate interior angles, since KN || LM and LN is a transversal).
3. LN = NL (Common side to both triangles).

Therefore, $\triangle KLN \cong \triangle MNL$ (Angle-Side-Angle (ASA) congruence criterion).

Since the triangles are congruent, their corresponding sides are equal:

• KL = NM (Corresponding sides of congruent triangles)
• KN = LM (Corresponding sides of congruent triangles)

Thus, the opposite sides of parallelogram KLMN are equal.