**Part(a) $35^{\circ};55^{\circ}$**

**Part(b) $35^{\circ};145^{\circ}$**

**Part(c) $35^{\circ};70^{\circ}$**

**Part(d) $35^{\circ};35^{\circ}$**

This question aims to find the pair of angles concurrent to the **sin x** and **cos y**.

**Congruent angles** are the angles that have the **same measure.** So all angles that have the same size will be called** congruent angles**. They are seen everywhere, such as in** equilateral triangles**, isosceles triangles, or when a t**ransversal intersects two parallel lines.**

In **mathematics**,** angles** that are equal in the measure are known as **congruent angles**. In other words, **equal angles** are also congruent angles denoted by the $≅$. They don’t point to the **same direction.** They don’t have to be on **lines of similar size.**

**Congruent Angle theorem**

There are **number of theorems based on congruent angles.**

**Vertical**angles theorem**Corresponding**angles theorem**Alternate**angles theorem**Congruent**supplements theorem**Congruent**complements theorem

**Vertical** **angles theorem**

According to the **vertical angle theorem**, vertical angles are always **congruent.**

**Corresponding** **angles theorem**

The **corresponding definition of angles** tells us that when two parallel line intersected to a third, the angles that have the same relative position at each point of intersection are known as **corresponding angles.**

**Alternate** **angles theorem**

When a **transversal intersects with the two parallel lines**, each pair of alternate angles is **congruent.**

**Congruent** **supplements theorem**

**Supplementary angles** are those whose sum is $180^{\circ}$. This theorem states that** angles supplementing the same angle are congruent angles**, whether adjacent angles or not.

**Congruent** **complements theorem**

**Supplementary angles** are those whose **sum** is $90^{\circ}$. This **theorem states** that angles that supplement the** same angle** are **congruent**, whether **adjacent or not.**

**Tips and tricks**

**Congruent angles**are just**another name for equal angles.**- All
**vertically opposite angles**are congruent angles. - All a
**lternate an**d corresponding angles formed by the**intersection of two parallel lines**and a**transversal are congruent.** - According to the
**definition of congruent angles**, “For any two angles to be congruent, they must have the**same size**.”

**Expert Answer**

**Step 1**

\[\cos(90-\theta)=\cos(90)\cos(\theta)+\sin(90)\sin(0)\]

\[\cos(90-\theta)=\sin(\theta)\]

**Step 2**

Using $\theta=35$ then,

\[\cos(90-35)=\sin(35)\]

\[\cos(55)=\sin(35)\]

\[35^{\circ},55^{\circ}\]

**Option $a$ is correct. $35^{\circ}$ and $55^{\circ}$ are the congruent angles to $\cos^{\circ}$ and $\sin^{\circ}$.**

**Numerical Result**

Option $a$ is correct. $35^{\circ}$ and $55^{\circ}$ are the **congruent angles** to $\cos^{\circ}$ and $\sin^{\circ}$.

**Example**

**Which pair of angles have congruent values for the $\sin x^{\circ}$ and the $\cos y^{\circ}$?**

**(a) $42^{\circ};42^{\circ}$**

**(b) $42^{\circ};48^{\circ}$**

**(c) $42^{\circ};138^{\circ}$**

**(d) $42^{\circ};132^{\circ}$**

**Solution**

\[\sin x=cos(90-x)\]

\[\sin(42)=cos(90-42)\]

\[sin(42)=cos(48)\]

**Option $b$ is correct.**

$42^{\circ}$ and $48^{\circ}$ are the **congruent angles** to $\cos^{\circ}$ and $\sin^{\circ}$.