Answer:
AC=3 in, m∠C=45° and BC=3√2 in.
Step-by-step explanation:
Given information: ABC is a right angled triangle AB=3 in. , m∠A=90° , and m∠B=45° .
According to the angle sum property the sum of interior angles of a triangle is 180°.
[tex]\angle A+\angle B+\angle C=180^{\circ}[/tex]
[tex]90^{\circ}+45^{\circ}+\angle C=180^{\circ}[/tex]
[tex]135^{\circ}+\angle C=180^{\circ}[/tex]
[tex]\angle C=180^{\circ}-135^{\circ}[/tex]
[tex]\angle C=45^{\circ}[/tex]
The measure of angle C is 45°.
In a right angled triangle,
[tex]\tan\theta=\frac{opposite}{adjacent}[/tex]
In triangle ABC,
[tex]\tan B=\frac{AC}{AB}[/tex]
[tex]\tan (45^{\circ})=\frac{AC}{3}[/tex]
[tex]1=\frac{AC}{3}[/tex]
Multiply both sides by 3.
[tex]3=AC[/tex]
The measure of AC is 3 in.
According to Pythagoras theorem,
[tex]hypotenuse^2=base^2+perpendicular^2[/tex]
[tex]BC^2=3^2+3^2[/tex]
[tex]BC^2=9+9[/tex]
[tex]BC^2=18[/tex]
Taking square root on both sides.
[tex]BC=\sqrt{18}[/tex]
[tex]BC=3\sqrt{2}[/tex]
Therefore, the missing measurements are AC=3 in, m∠C=45° and BC=3√2 in.
