To construct triangle ABC with the given specifications using a ruler and compass, follow these steps:
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Draw the Base: Start by drawing a line segment AB that measures 10.5 cm using the ruler. Label the endpoints A and B.
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Construct Angle ABC: At point B, you will need to construct an angle of 30 degrees.
a. Place the compass point on point B and draw an arc that intersects line segment AB on both sides.
b. Label the points where the arc intersects line segment AB as C (on the outside of the angle).
c. Without changing the width of the compass, place the compass point on the intersection of the arc with AB and measure a length that you will use to draw the 30-degree angle.
d. Draw the arc from point B to form an angle of 30 degrees above line segment AB.
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Construct Angle CAB: Now at point A, you need to construct an angle of 45 degrees.
a. Again use the compass to draw a larger arc that intersects line segment AB (continue from the previous step).
b. Set the compass to the same width and draw an arc from point A to intersect the previous arc. This will help in finding the intersection points for angle CAB.
c. Draw another line from point A through the intersection point of the arc at points C, thus creating the 45-degree angle at point A.
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Locate Point C: The intersection of the two angle lines from steps 2 and 3 will give you point C.
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Label the Triangle: Finally, label the vertices of the triangle. You should have triangle ABC constructed with AB = 10.5 cm, ∠ABC = 30 degrees, and ∠CAB = 45 degrees.
Note: The angles might need verification using a protractor, but this process will give you a triangle satisfying the angle and side conditions specified.
This triangle cannot actually exist according to the triangle angle sum property because if ∠ABC = 30 degrees and ∠CAB = 45 degrees, then ∠BCA = 180 - (30 + 45) = 105 degrees, which should be impossible with the given side lengths. Please double-check the angle specifications or side lengths as needed.