Angle Construction and Geometric Proofs
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10. The unknown angles
1° Draw an angle $\widehat{xAy}$ measuring $120^\circ$.
2° Draw its bisector $[Az)$.
3° Draw the bisectors $[At)$ and $[Au)$ of the angles $\widehat{xAz}$ and $\widehat{zAy}$.
4° What is the measure of each of the angles $\widehat{xAz}$? $\widehat{uAy}$? $\widehat{tAy}$?
14. Dynamic angles
1° Draw an angle $\widehat{xOz}$ measuring $56^\circ$ and a semi-line $[Oy)$ inside this angle such that $\widehat{xOy} = 26^\circ$.
2° Draw the semi-line $[Ou)$, such that the angles $\widehat{xOy}$ and $\widehat{xOu}$ are adjacent and equal.
3° Draw the semi-line $[Ov)$, such that the angles $\widehat{zOy}$ and $\widehat{zOv}$ are adjacent and equal.
4° What is the measure of the angle $\widehat{uOv}$?
17. Critical thinking
1° a) Draw two adjacent supplementary angles $\widehat{xOy}$ and $\widehat{yOz}$ such that $\widehat{xOy} = 40^\circ$.
b) Calculate $\widehat{yOz}$.
2° a) Construct, by using a ruler and a compass, the bisector $[Om)$ of $\widehat{yOz}$.
b) Calculate $\widehat{yOm}$.
3° a) Draw, inside the angle $\widehat{xOy}$, the semi-line $[On)$ perpendicular to $[Om)$.
b) Calculate $\widehat{nOy}$.
c) Show that $[On)$ is the bisector of $\widehat{xOy}$.
18. 1° a) Draw two adjacent complementary angles $\widehat{xOy}$ and $\widehat{yOz}$ such that $\widehat{xOy} = 30^\circ$.
b) Calculate $\widehat{yOz}$.
2° a) Draw the semi-line $[Ot)$ opposite to $[Ox)$ and the semi-line $[Or)$ opposite to $[Oy)$.
b) Calculate $\widehat{rOt}$, $\widehat{zOr}$ and $\widehat{rOx}$.
3° a) Draw, by using a ruler and a compass, the bisector $[Ou)$ of the angle $\widehat{xOy}$ and the bisector $[Ov)$ of the angle $\widehat{rOx}$.
b) Show that the semi-lines $[Ou)$ and $[Ov)$ are perpendicular.
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Step by Step Written Solution
Hi Matjar., let's solve this geometry problem together. We'll start with problem 17, which focuses on adjacent supplementary angles and geometric bisectors.
Problem 17: Critical Thinking Analysis
In part one, we are given that angles x O y and y O z are adjacent and supplementary. By definition, supplementary angles sum up to one hundred eighty degrees.
We are given that the measure of angle x O y is forty degrees. To find angle y O z, we subtract forty from one hundred eighty.
Calculating the difference, we find that angle y O z is equal to one hundred forty degrees.
Now let's visualize this with a drawing. We have a straight line for the supplementary angles and the bisector of y O z.
Geometric Construction
In part two, we construct the bisector O m of angle y O z. A bisector divides an angle into two equal parts.
To calculate angle y O m, we divide the measure of y O z by two. One hundred forty divided by two is seventy degrees.
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