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 transversal intersects two parallel lines.
Angle less than degree
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
Congruent angles
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 alternate and 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 $\sin^{\circ}$.
Vertical angle theorem
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 $\sin^{\circ}$.